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CHAPTER 4

Space–Dependent Properties of Neurons

4.1. Potential Fields of Single Neurons

A main source of data on the dynamics of neurons is the measurement of potential in the extracellular volume surrounding them. From measurement at n points in the volume v'(t, x, y, z), we infer the existence of distributions of current j(t, x, y, z) in the volume. Our interest lies in learning which neurons or parts of neurons sustain the emf generating the current, and how the values for the current relate to the active states of the generating neurons. Our basic approach is to measure a given field v’(t, x, y, z), postulate a time–varying distribution of emf constituting an active state o(t, x, y), predict the field v(t, x, y, z), and compare it to v’(t, x, y, z). If the degree of fit is unacceptable, we reject the model.

There are three main difficulties. First, there is only one field v for each model, but there are many possible models for each field v'. The acceptance of a model is not based only on goodness of fit, but on whether the geometry and parameters of the model conform in detail to the anatomy and physiology of the generating neurons or neuron parts. It is more difficult to develop and justify a model than to test it.

Second, the data required for testing proposed models must consist of measurements of potential at many locations in a neural mass. The measurements are made sequentially by moving an exploring electrode to each of a predetermined set of points for sampling, or preferably by simultaneous recording at many points. Each technique has advantages and limitations, and optimally both are used for any given mass. In practice the sheer bulk <Page 173> of the data and the need for concise display present formidable problems. Third, the labor of calculating predicted fields of potential by hand is prohibitive. Computers are required to perform the immense number of additions. There are several technical methods available for calculating the potential, including use of the dipole moment and solid angle (Woodbury, 1961); differencing (see footnote at the end of Section 4.1.2) to estimate source–sink density (Howland et al., 1955; Haberly & Shepherd, 1973); discrete networks approximating current vector fields (Rail & Shepherd, 1968); and the equivalent electrostatic field (Lorente de Nó, 1947b, Horowitz & Freeman, 1966). The electrostatic approach is rigorous, conceptually simple, and easy to apply with computer assistance. It is described and applied in the next sections. (For more general treatments, see the work of Plonsey, 1969 and Nicholson, 1973).

4.1.1. BASIS FUNCTIONS FOR MEASUREMENT OF POTENTIAL IN SPACE

The medium of the nervous system is water, which is divided by lipid barriers (the cell membranes) into a very large number of internal closed compartments (the cells) and an external compartment extending throughout the brain (the extracellular medium). The charge in this medium is composed of ions, not electrons. Because electrical measurements of potential do not distinguish types of ions, the amount of charge in a given volume of tissue is the algebraic sum of positive and negative ions appropriately weighted for valence. Except at cell membranes the sum in any prescribed volume is everywhere equal to zero. At cell membranes a deficit of negative charge is not distinguishable from an excess of positive charge by electrical measurement and by convention is equal to a net positive charge.

If the charge is distributed in space, then the charge dq in the differential volume of space dx dy dz at each point is the charge density

The presence of a fixed charge q is observed by placing a small test charge q₀ near it and measuring a force F acting on the test charge. For a distance η between q and q₀, by Coulomb's law,

where ε is a proportionality constant characteristic of the medium and i is a unit vector. The force is a vector in the direction of the line i joining q and q₀. A set of measurements of F in the space surrounding q delineates a field of force F(x, y, z). The intensity of the electrical field at each point E(x, y, z) is given by

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Work is required to move the test charge along lines of force in the field. For the field of force of q an increment of work dw is given by

where dr is an increment of distance along the line qq₀. From Eq. (2) and by integration from an infinite distance to η point at q₀,

The potential at η is

If we neglect dissipative forces such as friction, the value for the potential is independent of the path on which the test charge is moved. If the test charge is moved back to its starting point, the work is recovered. The field is conservative; therefore, the potential is a scalar function. The potentials due to multiple charges q_n in a field are additive:

The locus of points in space having the same value of potential is an isopotential surface. Just as a field of force can be described by lines of force, a field of potential can be described by a set of isopotential surfaces. The intersection of these surfaces with a plane used for illustration gives isopotential curves or lines. The isopotentials are continuous closed curves that do not cross. Lines of force cross isopotential surfaces only perpendicularly to them.

The values of potential in a field are measured with respect to the values at some reference level of potential. There is no unique value for potential at each point, because the values depend on the potential at a selected reference point. Ordinarily this point is conceived as lying at some great distance from the field of interest, so that the reference potential is v = 0, and at r = ∞, there is a zero isopotential surface. Any other point may suffice and may be required by technical considerations. In any case, measurements are made not of potential but of potential difference.

This poses a critical problem in the measurement of neural electrical activity. The fields to be measured may or may not extend throughout the brain, so that the reference point (location of the reference electrode) should ideally be placed on some other part of the body. In so doing, however, the potential fields generated by the heart, eyes, skeletal muscles, etc. may dominate the recordings and obscure the patterns sought. Therefore, it is often necessary to record potential differences in a neural field in which the reference potential is not equal to zero, i.e., both reference and exploring <Page 175> electrodes are deliberately placed in the field. This is known as bipolar recording. In other instances the attempt is made to place the reference electrode at the margin of the field of interest, e.g., low behind the ears or over bony cavities such as the frontal sinus. By convention (not by necessity) this is known as monopolar recording. Only when a reference point has been chosen is the potential function defined.

From the definition of potential the intensity of the electric field is related to the rate of change of potential with distance. The rate of change is zero along isopotentials and maximal in directions perpendicular to isopotentials. The maximal rate of change of potential in the vicinity of a point is the gradient. It is designated by Ñv and is given by

where i, j, and k are unit vectors in the x, y, and z directions. The field of force per unit charge is given by

This is a vector lying opposite to the direction of maximal increase in potential with distance. The gradient is positive uphill, whereas force is positive downhill.

The rate of change of the field of force in the vicinity of a point is designated by ÑE and is called the divergence. It is dependent on the charge density at the point. If the charge density is zero,

If the charge density is not zero,

Both the charge density and the divergence are scalar quantities. In relation to potential from Eq. (9), ÑE = –Ѳv and

At each point in an electrostatic field the divergence is proportional to the charge density (Poisson's equation)

In regions containing no charge it is equal to zero (Laplace's equation)

The solutions to Eqs. (13) and (14) for any ξ and for specified boundary conditions provide the basis functions for measurement of fields of potential.

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Example A. Consider two infinite planar sheets of positive and negative charge having uniform density, ξ = ± 1 . Let these lie parallel to the y and z axes at the values of x = ± c, so that the charge density is a function of x. In this case two of the partial derivatives are zero,

For points not in the two sheets, the charge density is zero,

The general solution is

Let us take for reference the value of potential at x = 0 as v₀ = 0, and designate the potential at x = c as v_c. From symmetry the potential at x = –c is –v_c, and for the space between the planes,

ac + b = v_c,                           –ac + b = –v_c.

Here b = 0 and a = v_c/c, so that

The potential is a linear function of distance from the origin between the planes. The gradient is

so that the field of force is uniform both in direction and magnitude between the planes. Outside the planes the boundary condition is that the potential not be infinite as x approaches infinity. Thereby a = 0 and b = ±v_c. The potential is uniform on each side, and is equal to +v_c for x > +c and to –v_c for x < –c. The gradient is everywhere zero, so there is no external field of force. §

Two infinite sheets of charge of opposite sign may be considered to form a hollow closed surface for which the radius of curvature is very large in comparison to the distance between the sheets 2c. In this case the negative side corresponds to the interior of a neuron at rest, and the double sheet is its lining membrane. Charge separation occurs only at the membrane, and the potential is constant throughout the interior, but it differs from the exterior by an amount dependent on the charge density function. If the reference potential is chosen as v = 0 at x = ∞ outside the surface, the potential inside the surface is v_m = 2v_c.

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4.1.2. BASIS FUNCTIONS FOR POTENTIAL IN CURRENT FIELDS

Up to this point charge has been considered as fixed in space. In the nervous system extracellular electrical fields arise only in conjunction with moving ions. By convention this current is said to flow in the direction of movement of net positive ions (cations), even if the actual flow is wholly owing to movement of negative ions (anions) in the opposite direction. In a bounded conductor the current is defined as the amount of net positive charge passing a complete cross section of the conductor per unit time

dq(coulombs)/dt(seconds) = i(amperes).

In a volume conductor the magnitude of current is defined in terms of the current di flowing through a specified surface area dA which is the current density

di(amperes)/dA(meters²) = j(amperes/meters²)

where j is a current vector perpendicular to dA.

Where does the moving charge come from (source) and go to (sink)? Because neurons can neither create nor destroy charge, there are no true sources and sinks in the brain. There are only two other possibilities. Either the charge is being pumped from one region to another, with changes in charge density dξ/dt in those regions, or it is moved in a closed loop. Neurons, however, cannot generate the immense electrical forces required to alter charge densities in aqueous solutions, except across membranes at which the voltage gradients are on the order of 10 million V/m. Even here the separation of charge does not change overall charge density, but only the local densities of anions and cations by separation of them. In the nervous system charge density ξ is constant at all times and places in the brain to within the smallest dimensions of a recording microelectrode. Therefore current flows only in closed loops.

The total current across any closed surface in the nervous system (such as a membrane) is always and at every instant zero. Current in must equal current out. These facts are expressed in mathematical form in terms of the spatial derivative or divergence of the current density Ñj. This is a scalar quantity expressing the rate of contraction or expansion of the flow at a point. The only two ways to increase the divergence are to add more charge or to decrease the density. In a source–free and sink–free volume such as the nervous system the divergence and the rate of change in charge density are related to each other by the equation of continuity,

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Because dξ/dt = 0,

Now we consider the relation between current, force, and potential. The movement of ions results from the application of force. As discussed in Chapter 3, electromotive forces of several kinds operate in the membranes of neurons, but in the volumes within and surrounding each neuron the only significant forces acting on ions are electrostatic and resistive. The mass of an ion is much greater than that of an electron, but in the frequency band characteristic of neural activity (0 – 10 kHz) there are no significant relaxation or inertial effects. Owing to the low magnetic permeability of water in comparison to its electrostatic permeability and to the low rates of change in flow, there are no significant magnetic effects. The ions do not interact viscously to produce vortices or eddy currents (such as occur in currents of water or air). All of these negatives mean that in the fields of force manifested as neural fields of potential, the ions move only in the direction of the lines of force. The rate of movement depends on the field intensity E and on the volume specific resistance ρ in units of ohm•cm²/cm (see footnote in Section 2.3.4). For each point the function relating j to E is described by Ohm's law,

The divergence is found by taking the spatial derivative of the current density at each point:

By Expansion,

in a homogeneous isotropic medium

From Section 4.1.1,

so from Eqs. (23)–(25)

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By Eq. (21), in a source–free and sink–free region,

This result is formally equivalent to Laplace's equation (14). In a region containing a source, by Eq. (20),

There is a formal equivalence of Eq. (28) to Poisson's equation (13) in which ρ is replaced by  and dξ/dt is replaced by ξ. The equivalence means that it is permissible to represent the lines of current in a homogeneous medium by lines of force in an electrostatic field. The sources and sinks at which the current lines start and end can be represented by equivalent positive and negative charge. What is not made clear is how this can be done in a medium without sources and sinks.

Moreover, Eq. (26) depends on two simplifications regarding the medium of the nervous system, first that it is homogeneous and isotropic, and second that it is purely resistive. It is clear that every current loop associated with a neural field must twice cross a high impedance, which is the membrane of the generating cell. In the near vicinity of an active neuron there are impedances imposed by the membranes of adjacent neurons. Beyond this near range (up to .3 mm) there are differences in the specific conductances of the grey matter, white matter, and the cerebrospinal fluid. Beyond the brain there are barriers imposed by the coverings (the dura and the skull) and ultimately by the surface of the head. None of these features is entirely negligible, although not all need to be introduced into the analysis of specific fields. The requirement, however, to introduce the membrane impedance discontinuity of generating cells is inescapable. That is, Ñρ^-1 cannot be zero.

The solution in every case requires the separation of each current field into two parts, one inside and one outside the neuron. The membrane becomes a boundary for both the inner and outer fields. An area of current outflow is a source for the outer field and a sink for the inner field. Obviously they are equal in total absolute magnitude. The reverse holds for an area of current inflow. For both the inner and the outer fields, the sum of fictitious sources equals the negative of the sum of fictitious sinks, because the total transmembrane current adds to zero. Both the impedance discontinuity and the operation of transmembrane electromotive forces are included in the boundary conditions already specified by the distribution of fictitious sources and sinks, which we will represent by j.

The set of fictitious sources and sinks (transmembrane current) is represented by an equivalent set of positive and negative charge, which we will <Page 180> represent by ξ or q. The potential at each point in the inner and outer fields is predicted by means of Eq. (7).^†

^† Equations (27) and (28) are the basis for the method of differencing as a means to determine the sources and sinks giving rise to a field of potential in a neural mass. Suppose that a set of measurements of potential v(X) is made in a three–dimensional lattice of recording sites X in a mass either simultaneously or at fixed latency with respect to a stimulus time. Each site X is at the center of a cube formed by the adjacent six sites X_n, n = 1, . . . , 6, lying each at the center of a face of the cube. The potential difference v(X_k) – v(X_n) between the center and each of the six sites is assumed to be proportional to the current flowing into the cube through the face of that site. Then in a medium with homogeneous specific resistance the net current inflow (sink density) or outflow (source density) at the center site is proportional to j(X_k) = S⁶_n=1 [v(X_k) – v(X_n)].

The method is conceptually simple, but in practice it is difficult to get measurements at sites along a number of parallel tracks sufficiently close together and in parallel planes sufficiently close together without damaging the tissue by electrode penetrations. The method tends to accentuate differences due to variation in background activity and time variance of the preparation on successive penetrations, as well as errors in the directions of successive penetrations which are never quite parallel. According to Eq. (27), the sum over all sites S_{all k}j(X_k) in the mass should be zero. Further, the results of the method must be checked by computing a potential field by means of Eq. (7) from j(X_k) and comparing it with the measured field for correspondence. In the author's experience the method does not give results approaching these conditions due to the above sources of error. Therefore, differencing yields hypotheses about j(X) and not definitive results. The anatomy of the mass provides an alternative source of hypotheses about j(X). This is the source used in Chapter 4. Irrespective of the source, however, Eq. (7) is required to test the hypotheses regarding j(X) against v'(X).

4.1.3. POTENTIAL FUNCTIONS FOR THE CORE CONDUCTOR

The core conductor (Section 2.3.4) represents a cylindrical membrane of infinite length and negligible diameter. We assume here that it has a nonuniform set of emf in its membrane at any instant of time. What are the expected functions of potential for the inner and outer fields?

The cylinder is conceived to lie along the x axis. The set of emf is given the values E_m(x) –1 for all x < 0, E_m(x) = 1 for all x > 0, and zero for x = 0. The parameters are constant and homogeneous: r_m and r_i (Section 2.3.4) are the membrane and internal specific resistances, and ρ is the volume specific resistance of the external medium.

The difference in emf in the two parts of the membrane establishes a longitudinal potential difference across the internal and external conducting media, so that longitudinal current flows to the right outside the cylinder and to the left inside the cylinder. The total longitudinal resistance from x = 0 to x = ±∞ is infinite, so that for x = ±∞ both the longitudinal current, i_e(x) and i_i(x), and the transmembrane current j_m(x) are zero.

The field of current is divided into an internal part and an external part.

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For any loop current path, the total resistance along the path in the external medium is much less than the total resistance along the internal path, so when the internal current field is considered, the external specific resistance is assumed to be ρ = 0. Therefore the potential is taken as zero everywhere outside the membrane. The potential across the membrane at any point except x = 0 should satisfy

The longitudinal current i_i(x) satisfies

for all x except x = 0. At x = ±∞, i_i(X) = 0. Therefore v_m(X) is given by

The internal longitudinal current from Eq. (29) is

The transmembrane current density is

These functions are illustrated in Fig. 4.1. For the internal field of current j_m(x) is represented by a set of fictitious continuously distributed sources and sinks. For the external field there exists a mirror image set

These are replaced by a set of equivalent charge densities.

For purposes of measurement, the continuous distribution of charge density is divided into N short segments of length ∆x, and a value for point charge is assigned to each value of x_i:

The electrostatic potential is calculated for a set of points (x, y, z), in the <Page 182> external medium by summation over the discrete charge:

FIG. 4.1. Relations between (a) potential, (b) longitudinal current (force), and (c) transmembrane current (source–sink density) for a core conductor in the steady state, in which the transmembrane emfs are uniform along x > 0 and x < 0 but are unequal at x = 0.

Example A. A set of isopotentials illustrating v(x, y, z) from Eqs. (31)–(39) is shown in Fig. 4.2a.  §

Example B. The procedure is repeated for the condition in which l is a variable, such that l(X) = 1 for x ≥ 0, and l(X) = 0.33 for x < 0. The resulting asymmetric function of v(x, y, z) in Fig. 4.2b is commonly observed for single neurons and sets of neurons. It results when the active membrane (E_m ≠ 0) <Page 183> constitutes most or all of the dendritic membrane, and the passive membrane (E_m = 0) is restricted to the axon, or vice versa. The dendrites have larger cross–sectional area, lower internal longitudinal resistance, and greater length constant than the axon. §

FIG. 4.2. Extracellular contours of potential (dark curves) and current lines (light curves) for a core conductor as specified in Fig. 4.1. (a) +λ = –l, corresponding to the idealized case of a neuron with symmetric active and passive membranes. (b) +l = –3l, corresponding to the case of the active dendritic tree operating into the axonal tree (at left). (c) +l = –9l, showing the effect of marked disparity in l, as in the case of the injury potential of a severed nerve, which is maximally negative at the cut end (at left).

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The pattern for v(x, y, z) in Fig. 4.2a occurs only when the parameters of the active and passive membranes are equal, such as when one of two comparable dendritic trees of a neuron is active and the other is passive.

Example C. l(x) = 1 for x > 0 and l(x) = 0.1 for x < 0. This pattern for v(x, y, z) in Fig. 4.2c is observed when an axon or a bundle of axons is cut at x = 0. For several minutes to an hour or more there is loop current due to inward diffusion of Na⁺ ions through the cut end and outward active transport of Na⁺ ions across the adjacent intact membrane. The potential with respect to a far distant point is known as the "injury" or "demarcation potential" and is everywhere negative. §

These examples illustrate a general principle. The sum of current sources and sinks across the membrane is zero, and the sum of equivalent positive and negative charge is zero. The charge density, however, in general is not uniform. If the source density is higher than the sink density, the field of potential is asymmetric, and the peak value of positive potential is greater than the negative of the peak value of negative potential, as shown in Figs. 4.2b and c. That is, total source = –(total sink), but in general, v_max ≠ –v_min.

Example D. If the core conductor is surrounded by a nonconducting medium, the external conducting medium is reduced to a thin cylindrical film. This corresponds to the experimental condition, in which an axon or nerve is excised from surrounding tissue and suspended in air or oil. In this case the radial derivatives, ∂v_m(y)/∂y = ∂v_m(z)/∂z are negligibly small and are set equal to zero, so that

The solution for v_e(x) in the external field is identical to the function of potential v_m(x), except that the sign is reversed and there is a difference in reference potential

where ε is a positive real number and v_o is the mean transmembrane potential at rest. §

More generally, the external potential function is a mirror image of the internal potential function, if the lines of current in both compartments are constrained parallel to the axis of the core conductor. This principle is an important feature of certain types of neural mass (see Section 4.3.5).

Equation (23) can be solved for the more general case in which ρ is nonuniform and is a function of space ρ(x, y, z) by appropriate methods <Page 185> (e.g., Nicholson, 1973). In the systems described here ρ is sufficiently uniform to be treated as a constant.

The advantage of the electrostatic equivalent model is that the potential can be evaluated at each point by scalar summation. If a computer is not available for simulation, the simplest alternative is to construct a three–dimensional network of discrete current lines to represent the vector field of current, and to solve the network equations for the potential difference between any two points.

4.1.4. POTENTIAL FIELDS OF AXONS

The core conductor model is applied to nerve axon for analysis of the distributions of both exogenous and endogenous currents, i.e., during both electrical stimulation and recording (Section 2.3.4 and 3.2.1). The analysis is simplified by suspending the nerve fiber or trunk in air or oil, which leaves only a thin sheath of conducting fluid around the nerve. This constrains the external field to a single dimension, so that the functions of potential, the gradient, and the longitudinal current outside the fiber are mirror images of those inside, except for the resting level for the potential with respect to a distant point (Example D, Section 4.1.3). For example, the action potential recorded monopolarly at the surface of an axon or nerve suspended in air is almost a mirror image of the intracellular potential except for scale and baseline. Over a straight and unbranched segment of axon or nerve the peak amplitude and velocity θ of the emf E(t – T) are constant, and T = x/θ. The field may be considered as stationary with respect to a set of coordinates undergoing translation at a velocity –θ. It is a moving dc field. As the field moves past a fixed recording electrode the potential as a function of time v_x(t) is directly proportional to potential as a function of distance v_t(X). The time and space derivatives are related by the velocity

In the frog sciatic nerve (Lorente de Nó, 1947b) a low–level stimulus is used, which activates only the largest axons having the highest conduction velocities. Higher–intensity stimulation activates smaller, slower–conducting axons, which leads to dispersion and to multiple crests in the extracellular recording. Both stimulating and recording electrodes are placed on a limited segment of nerve (e.g. 5 cm), which is twice as long as the wavelength of the compound action potential. Wavelength (28 mm) is the product of velocity (e.g., 28 m/sec) and duration (e.g., 1 msec). It is necessary to KIII or inactivate the terminal end of the nerve trunk in order to avoid the appearance of two partly overlapping action potentials, the second being inverted, due to recording of the traveling wave at the terminal end with the reference electrode.

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This so–called "monopolar" recording between an intact and an inactivated membrane displays the effects not only of changes in transmembrane emf but also of intramembranal longitudinal emf between the two electrodes. These secondary events called "longitudinal polarization" by Lorente de No (1947b) give rise to some of the multiple deflections (ß, γ, ε) seen to follow the main spike (Aα) in the compound action potential (Example A, Section 2.3.1).

This artifact is avoided by recording bipolarly between two electrodes placed close together on the nerve in comparison to the wavelength of the action potential. This gives the difference in potential ∆v between the two points separated by ∆x, varying as a function of time. The bipolar record is treated as proportional to the gradient ∂v_e/∂x by Eq. (42). The potential v_e as a function of time (or distance) is obtained by graphic integration. The slope gives the divergence ∂²v_e/∂x², which is proportional to transmembrane current density (Fig. 4.3).

FIG. 4.3. The external compound action potential (–v_e) of the frog nerve (alpha fibers) and (b, c) its first two derivatives (Lorente de Nó. 1947b).

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This empirical curve ∂²v_e/∂x² is used to specify the values of a set of 26 point charges fixed on a line segment. From these values, the external field for the action potential is calculated using Coulomb's law (Fig. 4.4). The value of potential along each line parallel to the axon at a distance r is the potential as a function of distance and is proportional to the potential as a function of negative time. These are the basis functions to predict the waveforms recorded by electrodes spaced at a distance η from the center of the nerve trunk (Fig. 4.5).

FIG. 4.4. Field of the compound action potential and lines of current in the external medium. The diagrams on top show the experimental arrangement, where n is the nerve, cm the conducting medium, and ins the insulating material (Lorente de Nó. 1947b).

Experimental verification is done by placing the nerve on a piece of blotting paper soaked with normal saline and recording action potentials at different distances from the nerve. Although the field is calculated (Fig. 4.5) for a volume and the measurements are made in a plane, agreement is satisfactory (Fig. 4.6).

Comparison of the extracellularly derived action potential –v_e in Fig. 4.3, with the action potential recorded intracellularly (Fig. 3.5b) shows that the two observed events are similar, except for the difference in sign of potential. The similarity is explained in Example D in Section 4.1.3. Comparison of the curve –∂²v_e/∂x² in Fig. 4.3c with action potentials recorded extracellularly in a conducting medium (Figs. 2.13 and 4.7) shows similarity in waveforms with reversal of sign. Both events are triphasic. From Eq. (147) in Chapter 2 the second spatial derivative of potential is proportional to transmembrane current density. Therefore, the monopolar extracellular recording of an action potential in tissue reflects the change in transmembrane current density as the action potential travels past the recording electrode, whereas the monopolar extracellular recording in air reflects the <Page 188> change in transmembrane potential. Neither of these two relations is an exact correspondence.

Fig. 4.5. Theoretical action potentials of nerve in a conducting medium at points (a) 2.4, (b) 4, (c) 8, and (d) 16 mm from the nerve axis (see Fig. 2.13) (Lorente de Nó. 1947b).

The triphasic wave form is seen characteristically at the midpoint of a long nerve fiber. Positivity associated with current outflow occurs at the foot and tail of the axon spike; the intervening negativity is associated with current inflow and with the region of active membrane. However, at the site of initiation of an action potential, e.g., in the initial segment, the wave form lacks the initial positive component; it is diphasic negative–positive. At the site of termination of the axon the second positive component is lacking; the wave form is again diphasic but positive–negative (Fig. 4.6). A theoretical basis for these facts is given by Lorente de Nó (1947b).

4.1.5. NODES AND BRANCHED FIBERS

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FIG. 4.6. Recorded action potentials at the indicated points in the medium. The volley enters the conducting medium at x = 0 and leaves it at x = 26. At both points the wave form is diphasic. Between them it is triphasic. The field in Fig. 4.4 corresponds to the instant labeled 2 (Lorente de Nó. 1947b).

FIG. 4.7. Action currents recorded between the points shown at the right in respect to the positions of nodes of Ranvier of a single myelinated axon, demonstrating saltatory conduction (Huxley & Stämpfli, 1949).

The picture of the action potential as a smooth wave gliding unchanging along an endless conductor is a useful fiction that holds for large nonmyelinated axons and for large bundles, but not for most neurons. They are <Page 190> shaped like bushes and trees, not like straws. Excitation which has begun at some section of the membrane develops and dies out with some characteristic time course. A field of current grows and collapses in a form determined by the geometry of the local cylindrical system. At the height of intensity of this field, excitation begins at one or more other parts of the tree, often relatively far distant. New, different, and partly overlapping fields of current arise. These are conditioned by differing but interconnected geometries. At the new crest, still another generation of fields begins to grow. The potential recorded in the vicinity of a neuron with respect to some distant point is the sum of all these successive and time–varying events, weighted according to distance from the active membranes.

An example of this is the sequential activity of nodes of Ranvier in single myelinated axons dissected free from nerve trunks. Active membranes during electrical activity of such axons occur only at nodes, and agents such as cocaine, sodium–free solutions, cold, ultraviolet radiation, etc., which block conduction, do so only when applied to nodes and not to internodes. The excitation of a node is achieved by an outward current across the nodal membrane, which is established either by an exogenous stimulating current or by activation of a nearby node. This is followed by an intense but brief inward flow and a secondary outward flow, when secondary activation of an adjacent node has occurred. At internodal segments there is only outward flow. The crest of the inward flow occurs at progressively later times for nodes at greater distances from the site of initial activation (Fig. 4.7 from Huxley & Stämpfli, 1949).

The nodes of Ranvier are located about 2 mm apart in each of the larger axons of the frog sciatic nerve, but they are randomly spaced with respect to nodes in other axons in the nerve. The record of activity by a single electrical stimulus applied to the nerve trunk is the smoothed average of successive activity at many nodes. The average transmembrane current density and the average transmembrane potential estimated by recording in air are consistent with each other, but they do not contain information about the local distributions and intensities of currents of single fibers.

The application of core conductor theory (Section 2.3.4) to the branched dendritic trees of central neurons is largely a problem in geometry, together with the search for neuroanatomical correlates. Rail (1959; 1968) provides an extensive set of solutions for differing types of neuron geometry, particularly for varying degrees of branching. Explicit parameters are given for the number of primary dendritic trunks (treated as cylinders) extending from the soma (treated as a sphere), the distance to the first branch point, diameter, degree of taper (if any), and length constant, the number of secondary branches and their parameters, tertiary branches, and so forth.

For a particular degree of branching, which appears to lie in an <Page 191> intermediate range between the extremes of paucity and profuseness of dendritic branching revealed by Golgi staining techniques, the entire tree can be described as if it were a single unbranched cylinder, in so far as its electrical relationships with the soma are concerned (Fig. 4.8). Distances x along neuron branches can be expressed in units of the length constant l so that electrotonic length X is given by

FIG. 4.8. Reduction of a single branched dendritic tree (S_id_i³/² = const) to an equivalent cylindrical core conductor (Rail, 1962).

The length X can be expected to vary with distance along a branching system, even if the specific resistivities of the membrane and of the inner medium are homogeneous, because as the branch diameter decreases, the surface area diminishes proportionately, but the cross–sectional area decreases as the square of the diameter. Rall (1962) has defined a "generalized electrotonic distance" as

For the steady state the equation for the core conductor may be written as

<Page 192>

FIG. 4.9. Reduction of neuron geometry to a spherical soma and a set of radiating cylinders representing multiple dendritic trees (Rall, 1962).

FIG. 4.10. Cross section through the field of potential of the neuron schematized in Figs. 4.8 and 4.9, in which the soma provides the sink and the dendritic trees provide the current source (see Fig. 4.2) (Rall, 1962).

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for that case in which dendritic branching satisfies the relation

Here the sum of the diameters d_j, each raised to the power, of all outgoing branches at the jth branch point having the electrotonic distance Z from the soma must equal the diameter of the main trunk d₀ to the power times e_KZ. The constant K represents the degree of branching. When K = 0, the system is equivalent to an unbranched cylinder of infinite length (Fig. 4.8). Negative and positive values for K denote lesser or greater profuseness of branching. For K = ∞ the tree is equivalent to a cut end, and for K = –∞ the system is equivalent to a sealed end.

An example is shown in Fig. 4.9 of a neuron having seven main dendrites regarded as infinite cylinders extending from a spherical soma. The soma is uniformly depolarized and constitutes the sink; the dendrites provide the sources. The distribution of sources and sinks is calculated by the core conductor equation applied to the internal field. The external field of potential (Fig. 4.10) calculated from the mirror distribution resembles that in Fig. 4.2c.

4.2. Potential Fields of Neural Masses

4.2.1. MEASUREMENT OF OBSERVED FIELDS

The active states of neural masses are manifested extracellularly at various recording sites by time–varying spatial distributions of action potentials and of dendritic potential. If a microelectrode is inserted into a neural mass at a point, the amplified potential with respect to a distant site (the monopolar recording) displayed with an oscilloscope shows many action potentials on each sweep, with one or more repeated action potentials at relatively high amplitudes and many others at amplitudes diminishing to the level of thermal noise in the electrode tip. The action potentials are superimposed on relatively slowly varying potentials. The amplified potential may be passed through a high–pass fitter to remove the slower waves (e.g., below 300 Hz) and a low–pass fitter to remove the action potentials (e.g., potentials changing at rates above 300 Hz).

The record of action potentials is passed through a threshold comparator, which emits a 1–V 1–msec rectangular pulse each time an action potential occurs with an amplitude above a preset value. The output of the comparator is a pulse train p'(t, x, y, z), which represents the time of occurrence of pulses of one neuron or a set of neurons in the mass. The distance between the active neuron or neurons is usually unknown, but the action potential of a neuron is often still detectable when the electrode tip is moved <Page 194> a distance of up to 100 µm from the site of maximal amplitude. The distance over which a neuron contributes to one recording p'(t,x₁,y₁,z₁) is < 100 µm. A set of measurements made simultaneously with multiple microelectrodes at a set of n points yields an activity distribution in the pulse mode for the mass p'(t, x, y, z), which is to be used to estimate o_µp(t, x, y) for the µth set in the mass.

The output of the low–pass fitter represents the sum of potentials, which is established at the recording site by dendritic currents of neurons in the mass, together with possible contributions by glia and by the network of blood vessels in the tissue. The contributions of single neurons cannot be distinguished, and there is no general rule to determine over what distances neurons in the mass may contribute to the potential at any point. A set of measurements of the extracellular low–frequency potentials at a set of n points gives a neural activity distribution in the wave mode v'(t, x, y, z), which is to be used as the basis for estimating o_µv(t, x, y) for the µth set in the <Page 195> mass. The set of n points is commonly restricted to the surface of the mass, to a plane, or a line transecting the mass, etc. The symbols used to denote restricted sets of values of potential are listed in Table 4.1.

Because of the time variance of the active state at each point, ideally the measurements at the set of n points are made simultaneously from n electrodes. In practice the technique is limited to use with arrays of electrodes at the surface of a neural mass. In depth recording the brain does not withstand placement of the large numbers of electrodes required, because the blood supply is impaired and many of the afferent and efferent axons are transected. For multichannel recording the number of electrodes is limited by the number of available amplifiers, usually 16 ≤ n ≤ 100. The spacing of electrodes is based on the anticipated surface "grain" or maximal spatial frequency of v'_z(t, x, y) and ranges from 100 µm on the cortex to several centimeters at the surface of the head. The display is in the form of a set of isopotentials in a projection plane v'_t,z(x, y), which are found by second order extrapolation and curve–fitting to the measured values at each successive time t of observation.

The alternative to simultaneous multichannel recording is serial penetration of the volume with a single electrode. Stable active states must exist without change in potential for the duration of the sampling procedure at n points. This holds for very slowly changing events, such as the injury current of nerve and muscle, but the potential in most fields of physiological interest changes rapidly. The solution to this problem requires the use of impulse input and the analysis of functions of potential at a set of points with poststimulus time T.

The neurons of an intact neural mass usually generate fields of potential in response to electrical stimulation of an appropriate afferent pathway. If the impulse response is invariant over a large number of trials, then the function for potential at any one point v'(T, x, y, z), can be stored, the electrode moved to a new point and a new sequence recorded, stored, and so forth, at a set of points coextensive with the volume of the potential distribution.

In using this technique it is essential to place a monitor electrode at a fixed central point in the neural mass and to record at each point x, y, z and at the monitor on each trial in order to demonstrate the stability of the impulse response. If the impulse response varies from trial to trial at any one point, ensemble averages or AEPs are taken at the recording and monitor sites (T, x, y, z). The ensemble average at the monitor site must be invariant, showing that the function of the neural mass is stationary.

The display of the depth potential function  (T, x, y, z) is in a set of planes for fixed values of T and x, in which isopotential curves are fitted to the data. The number of electrode tracks in the y dimension is limited by the number of penetrations the neural mass can sustain without deterioration <Page 196> and is commonly 5–10. The number of measuring sites on each penetration in the z direction depends on the complexity of the field and is usually ≥ 10.

The number of planes in the x direction is commonly 3–5, although in conditions of axial symmetry as in the spinal cord and some areas of cortex, 1 may suffice. The number of time intervals at which potential is measured in the function (T, x, y, z) is commonly 100, so that the number of measurements of  for each field is on the order of 10⁴ times the number of planes in x.

4.2.2. BASIS FUNCTIONS FOR POTENTIAL FIELDS OF NEURAL MASSES

The function (T, X), where X is x, y, z (Table 4.1), represents the potential field of the impulse response of the neural mass. The aim of analysis is to determine from the potential field one or more of the active states of the mass. Each mass is composed of one or more KO and KI sets, and the active state variables are defined only for those sets (Section 1.3.2). We must determine which KO and KI sets in a neural mass contribute to the function

(T, X). Because the fields of potential are superimposed, we infer that the potential is the sum of the fields generated by M KO and KI sets in the mass. The procedure for analysis consists in postulating M source–sink distributions, calculating v(T, X), testing it against (T, X), and inferring the set of active states from the acceptable source–sink distributions.

The current sources and sinks in a neural set µ are almost always fixed in space, because the neurons are fixed, so they are represented by an array of equivalent fixed point sources and sinks q_µ(X). The amplitudes of sources and sinks depend on time in two ways. First, when an event is initiated, such as an action potential at a node of Ranvier or a PSP at a synapse, the amplitude changes as a characteristic function of time f(T). Second, when an action potential propagates along an axon, there is a succession of virtually identical events at nodes of Ranvier, but there is a delay Ta in the onset of the event at each node. The amplitude is f(T – T_a), where f(T – T_a) = 0 for T < T_a. When a neural set is activated by a volley in a compound afferent nerve, each local subset is activated in a sequence. The sequence depends on the location of each subset in q_µ at X_n with respect to a reference point in the set at X₀, and the delay in each direction T_a(X), with respect to stimulus time T = 0. That is, the afferent tract provides delay and translation. These operations can be expressed by an operator

By this definition of T_n the function f_µ(T – T_n) is specified in terms of poststimulus time and location in the set, where n is a triple index, n = i, j, k in the x_i, y_j, z_k coordinates.

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For the source–sink density of a subset in set µ, which is q_µ(X_n), the amplitude as a function of time and location is given by the product q_µ(X_n)f_µ(T – T_a). The predicted potential field for the set is calculated by summation

The function v_µ is a spatial basis function. The sum of basis functions over µ = 1, 2, . . . , M yields the predicted potential field which is tested against the observed field by

where ε is an error function.

According to Eqs. (47)–(49), there are four requirements for constructing each basis function:

(a) The source–sink distribution in a local subset of neurons in a set µ is reduced to a charge module representing the cellular architecture, such as a pair of source–sink spheres. The charge sum over the module must be zero.

(b) The locations of the modules in space are specified by a geometry representing the set, such as a segment of a plane or cylinder, denoted q_µ(X).

(c) The sequence of activation for the array of modules T_n, representing input delay and translation, is postulated. In most instances the input volley can be treated as having constant velocity 9 in one direction. The X direction is usually chosen as the axis of input translation, so that T_x(X) = x/θ, T_x(y) = 0, and T_z(z) = 0 in Eq. (47). For equal increments of ∆x there are proportionate increments of ∆T_x.

(d) A time–dependent function f_µ(T – T_n) is assigned to every subset, which differs over the set only in time of onset T_n.

It is essential to recognize that afferent delays in activation over a neural mass often give rise to the appearance of traveling waves of potential, but the waves are not propagated or conducted, nor are they reflected at the boundaries of the mass. The wave equations of classical physics are not applicable. The waves result from the finite velocity of the input channel. The delays must be accounted for in the measuring process, even though their possible functional significance is obscure.

In many cases the input delay is negligible, and the value for T_x can be set equal to zero by appropriate choice of T_n or f_µ(T). Equation (48) is then <Page 198> modified as

A new function is defined

Therefore,

Each component field is represented as having a characteristic spatial distribution, and each has a characteristic weighting function to describe its amplitude over time. This formulation is used here extensively for the global description of fields of neural masses.

Example A. The choice for the local source–sink geometry depends on the typical structure of the generating neurons. If the afferent axons form synapses on one of two dendritic trees, which extend on opposite sides of the soma (often called bipolar neurons), the source and sink for each neuron can be represented as a function of distance z along the geometric axis of the neuron. Let us represent each source by a unit of charge density ξ and each sink by –ξ. We assume that the sources for a set of neurons form a homogeneous sphere of unit radius, a = 1, with its center at x = 0, y = 0, z = 1, and the sinks form a homogeneous sphere of unit radius centered at x = 0, y = 0, z = –1. We assume that f(T) is constant, and that T is zero across the set.

The potential v⁺(X) for a spherical distribution of positive charge outside the sphere is calculated for v(z)

The potential inside the sphere is

The same functions hold for v^–(X). The potential for the set is given by v_T,x,y(z) = v⁺_T,x,y(z) + v^–_T,x,y(z) and is shown in Fig. 4.11. The potential function is that of a distributed dipole field.

If the distance z » 2a, then the distributed dipole can be treated as a point dipole at x = 0, in which the dipole moment µ is a vector given by the product of the total charge in one pole Q and the distance of separation

<Page 199>

FIG. 4.11. Potential as a function of distance along a line through two homogeneous distributions of charge representing a source and a sink with equal current density. This is a form of the distributed dipole field.

2a, where a is a vector. The potential is

where φ is the angle between µ and the z axis and η is the distance between the origin and a point n. Use of Eq. (57) to predict the field of a neural mass requires vector summation. Scalar summation over pairs of point charges is simpler. §

Example B. If each neuron in the local subset has dendrites radiating in all directions from the soma (often called stellate neurons), all of which are activated so that the soma forms the source and the dendrites form the sink, a different field of potential results. We assume that the source for the subset is represented by a homogeneous sphere of radius a = I , centered at x = 0. The sink forms a concentric hollow sphere of outer radius b = 3 and inner radius a = 1, also centered at x = 0. The charge density of the source is greater than the charge density of the sink by –ξ^– = (b³–a³)/a³, so that ξ⁺ = –26ξ^– from the ratio of volumes. Thus ƒ(T) is constant and T = 0. The potential for the source v⁺(η) is given by Eqs. (55) and (56). Inside the hollow sphere of the sink,

Between the inner and outer radii,

<Page 200>

Outside the sphere,

The potential as a function of distance from the center is v(η) = v⁺(η) + v^–(η) as shown in Fig. 4.12 (see also Figs. 4.2c and 4.10).

FIG. 4.12. Potential as a function of distance along a line through the center of two concentric homogeneous distributions of charges representing a source and a sink of unequal current density. This is a form of the closed field or monopole field (from Biedenbach & Freeman, 1964).

This field differs in three ways from the distributed dipole field. First, the maximum and minimum values of potential are asymmetric. In the case illustrated there is virtually immeasurable negative potential even at the outer bound of the region of sink. The poles in the dipole field are equal but opposite in sign of potential. Second, the locus of reversal in sign of potential, which is the zero isopotential surface, does not have the same location as the locus of reversal in sign of charge, which is the locus for reversal of sign of transmembrane current. These two loci are identical within the distribution <Page 201> of sources and sinks for the dipole field. Third, the potential is zero everywhere outside the anatomical distribution of the filaments of the generating cells, whereas the potential of the dipole field can be recorded at great distances from the generating neurons. This feature led Lorente de Nó (1947a) to apply the descriptive label "closed field" to the field of concentric charge distributions. §

Example C. Suppose that the distributed source for a set of neurons is randomly distributed about a center at x = 0, with unit standard deviation, so that

The potential is given by

and is shown as the solid curve in Fig. 4.13. Let ξ(η) be approximated by a homogeneous sphere of charge with a radius of 2 standard deviations and centered at x = 0. The approximating potential from Eqs. (55) and (56) is shown by the dashed curve in Fig. 4.13. The difference between the predicted and approximating potentials is at the level of error ε in measuring (T,X). §

FIG. 4.13. Potential as a function of a radial distance from the center of a spherical homogeneous charge distribution, and from the center of a charge cloud decreasing in density according to the normal distribution in all directions. The potential function is insensitive to the details of the charge (source sink) distribution. See text for discussion.

Figure 4.13 in Example C demonstrates the powerful smoothing that occurs when the potentials are added across an array of distributed charge. <Page 202> The details of source–sink distributions in local subsets are not reflected in (T, X) so the geometric configuration of charge representing local subsets should be given the simplest form possible. This is why the dipole field and the closed field are the two most useful basis functions for describing v(X). Because of the insensitivity of the potential function v(X) to details in the equivalent charge distribution q(X), the necessary parameters for q(X) are merely the locations of the centers of each distribution of positive and negative charge, and the spread of each distribution.

Isolation and identification of each component f_µ(T)v_µ(X), µ = 1, 2,..., M, are based on one or more of several conditions that may hold for each given event. The delay T_µ for the onset of each component may vary. This is the basis on which the components of the compound action potential are identified. The rates of change for ƒ_µ(T) may differ by one or more log units. On this basis the presynaptic and postsynaptic components of an evoked potential are identified. The functions v_µ(X) may be distinguished by the use of two inputs, one input being given to one of two overlapping KO or KI masses and the second input to the other or both masses. Although there is no general solution to the problem, the necessary information includes the topology and geometry of the masses and a prediction of the form of the output. This information is embodied in the set of temporal and spatial basis functions, f_µ(T) and v_µ(X).

4.2.3. COMPOUND POTENTIAL FIELDS: MODULAR ANALYSIS

On the basis of cell geometry neurons having an axial geometry (e.g., bipolar neurons, Golgi type I neurons such as cortical pyramidal cells having long apical dendrites and long axons, etc.) can be expected to generate predominantly extracellular dipole fields, whereas neurons having a radial geometry (e.g., Golgi type II neurons such as cortical granule or stellate cells or cells with short axons) should generate predominantly extracellular closed fields. Owing to the irregularities of distribution in the branches of neurons and in active sites in the membrane, neuron fields do not conform precisely to these idealized models. When the approximation fails, they can be regarded as the sum of a dipole and a closed field. This becomes important in considering the fields of KO and KI sets in neural masses. The local details of the separate fields of participating neurons are not discernible, because their branches are so densely interwoven, and the equivalent external charge density function approximates a continuous distribution over each set rather than over each neuron. <Page 203> The form of the resultant multicellular field depends as much on the geometry of the set as on the geometry of the single neurons.

Let us suppose that the field of each neuron can be represented by a vector quantity expressing the dipole component [Eq. (57)] and a scalar quantity expressing the closed component. Two cases arise. If the axes of the neurons in a pool are randomly directed, then the vector components cancel and leave only the scalar components. The field of the set is closed. Thus the field of reticular and nuclear structures are characteristically low in amplitude, localized, and lacking in a well–defined "turn–over" or zero isopotential surface. If the axes of the neurons lie parallel to one another, then the vector components add to produce a vector or dipole field for the set. The scalar components add also, so that the observed field may be somewhat asymmetric. The dipole field is characteristic of the laminar populations in the cortex, which generate high–amplitude fields having a well–defined zero isopotential surface in the midregion of the cortex, and which can be detected outwardly as far as the scalp and inwardly through much of the brain. The cortex also generates nondipole fields (e.g., the field of the "recruiting" response). It is never safe to assume that the locus of the "turnover" in a cortical field of potential corresponds to the locus of membrane current reversal for the neurons generating the field, even when the set of neurons is plane or nearly plane.

In most parts of the neocortex the neurons have a common orientation but are displaced vertically from one another in the laminar tissue. In the simpler and phylogenetically older parts of the cortex (paleocortex) such as the hippocampus and prepyriform cortex, the pyramidal neurons share not only an orientation but a common depth of their somas as well. This gives these structures their striking laminar appearance in histological cross sections. These and similar structures generate potential fields having the highest amplitudes to be found for spontaneous and evoked potentials anywhere in the brain, particularly in comparison to those from reticular and nuclear masses (see Section 7.3.3).

A flat sheet of axially symmetric and parallel cells generates a plane dipole field. The layers of neurons in the nervous system usually occur in the form of curved surfaces. The effects of curvature are to increase the amplitude of potential in the concavity, to decrease the amplitude over the convexity, and to shift the zero isopotential outwardly. This inequality in amplitudes is found in the fields generated by the hippocampus, the olfactory bulb, the lateral geniculate nucleus, the superior colliculus, etc., all of which are curved surfaces. The disparity becomes marked as the radius of curvature approximates the length of the component cells. In the case where the curved layer is virtually in the form of a sphere, as in several brainstem nuclei (Lorente de Nó, 1947a), a closed field results.

Example A. The superior olivary complex lies in the brainstem and is part of the auditory system. The two main structures are the S segment, <Page 204> which receives afferent axons from the ipsilateral cochlear nucleus and the medial segment, which receives afferent axons on the lateral side from the ipsilateral cochlear nucleus and on the medial side from the contralateral cochlear nucleus. The bipolar neurons of each segment form a single layer with dendritic branches on both sides of the layer. The medial segment is a plane, and the S segment is strongly curved. The current source and sink of each local subset for each segment are represented by an equivalent charge dipole. The medial segment is represented by a plane set of dipoles, and the s segment is represented by two spherical dipole arrays.

FIG. 4.14. Click–evoked potentials recorded from various points in the superior olivary complex of the cat (Biedenbach & Freeman, 1964).

<Page 205>

A click delivered to the contralateral ear results in an evoked potential. The ensemble average _Xc(T) or AEP is nonoscillatory (Fig. 4.14) and closely resembles a PSP. There is no significant delay across the sets of neurons. The potential function is evaluated only at the crest of the response. The spatial distribution of potential T_c(X) has the form of a symmetric dipole field (Fig. 4.15) with the zero isopotential surface lying at the medial edge of the medial segment and with negativity on the medial side. An ipsilateral click results also in a nonoscillatory AEP. The function _Xi(T) is very similar to _Xc(T). The field of potential T_i(X) is an asymmetric dipole field, negative laterally, with the zero isopotential surface at the lateral edge (Fig. 4.16). The amplitude of the negative pole is higher than the inverse amplitude of the positive pole, and its volume extends further dorsolaterally to include the s segment. The structure of the S segment implies that on activation it would produce a closed field partially overlapped by the field of the medial segment. It is known from unit recordings that its neurons are activated by clicks. It is concluded that the ipsilateral impulse response of the superior olivary complex _i(T, X) is the sum of a closed field generated by the S segment v₁(X) as in Fig. 4.12, and a dipole field generated by the medial segment v₂(X) as in Fig. 4.11. Therefore, if both have the same time function f(T) = _Xi(T),

In general the fields of potential of two or more sets contributing to the field of a neural mass overlap in the volume of the mass, and separation by use of two or more inputs is not feasible. For each of the planes of measurement, y, z, transecting the mass, the values for potential _x(T, y, z) form a two–dimensional matrix consisting of the values in time at each point _x(T), and the values at each time over the set of points T(X) We can assume that the neurons in each set p have local and global geometries differing from those in other sets. The functions v_µ(X) must be different, and the functions ƒ_µ(T) may or may not differ significantly. The contribution of any set v_µ(T, X) over a set of points is uniquely different from that of any other set.

If the delay in onset of an active state through the mass is negligibly small in comparison to its duration, the value for the function of amplitude with time f_µ(T) is the same for all local subsets in each set µ at each T. In this case the contributions to the potential of the nth set at any point in the mass can be treated as the product of a factor v_µ(T) and a linear coupling coefficient for the set to that point l_nµ. For any point, the sum of contributions is

<Page 206>

FIG. 4.15. Fields of potential at the crest of the response to contralateral click stimulation of the medial accessory nucleus of the superior olivary complex. P5 and P7 designate the distance of coronal section posterior to the midaural plane (Biedenbach & Freeman, 1964).

<Page 207>

FIG. 4.16. Field of potential at the crest of the response to ipsilateral click stimulation of the S segment and medial accessory nucleus. P3.5, P5, P6, and P7 designate the distance of coronal section posterior to the midaural plane (Biedenbach & Freeman, 1964).

where _n is the mean potential at n over time and ε_nT an error term. In this way the potential function is represented by a set of n linear equations, in which the p time functions of the sets are the factors, and the n recording sites play the role of a stratifying variable.

If the µ factors are statistically independent of each other, the appropriate procedure is factor analysis of _x(T, y, z) to determine the principal components of the variance, which are estimates of the factors ƒ_µ(T). The values for λ_nµ are determined by linear regression over the set of n equations in the form of Eq. (64). The values are treated as estimates for v_µ(X), from which contour maps are constructed.

The condition of statistical independence seldom holds. The time functions of different sets in the same mass may be similar, because the active states of the sets are coupled by interaction. If there is no coupling, the time <Page 208> functions may be similar because the properties of the neurons in the two sets are similar. In this case the factors f_µ(T) cannot be estimated from the principal components of the variance. By inspection of the data _x(T, y, z), two or more subsets of points are selected at which the AEPs reflect the main features characteristic of the field of the mass. The ensemble average or AAEP over each subset is used to estimate the factors f_µ(T) and the coupling coefficients λ _nµ, are determined as before.

FIG. 4.17. Fields of potential recorded in the anterior septum and accumbens nucleus in the cat on electrical stimulation of the posterior septum (a) (Freeman & Patel, 1968).

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FIG. 4.18. AEPs recorded at points on the diagonal line shown in Fig. 4.17d. The arrows show the times at which the field maps were taken (Freeman & Patel, 1968).

<Page 210>

Example B. A single–shock electrical stimulus is given to a point in the septum of the cat (Fig. 4.17a), and a set of AEPs is taken from a coronal (vertical) plane yz, 1.5 mm anterior to the stimulus site. A representative electrode track, y = constant, is shown in Fig. 4.17d on which the crossmarks indicate the depth of recording z. A set of isopotentials v_T,x(y, z) representing _T,x(y, z) is shown in Fig. 4.17 for several times T after the onset of the event. A representative set of values for _x,y(T, z) is shown in Fig. 4.18. AAEPs averaged over selected sites at maxima in the field of potential of the mass are shown in Fig. 4.19.

The analysis is based on the assumptions that resistivity p is constant; the number of sets p is 2; the delay in onset T is negligible in each set; the time function for each set f_µ(T) is represented by ensemble averages of AEPs at points near the center of each set, but relatively distant from the centers of other sets; and the fields of the sets are superimposed. By linear regression using Eq. (64) two sets of λ_nµ are found from which two sets of isopotentials are constructed (Fig. 4.20).

FIG. 4.19. Estimates of the time course of the active states in each of the two neural masses, based on selective averaging (Freeman & Patel, 1968).

<Page 211>

FIG. 4.20. Characterization of two overlapping fields of potential (a dipole field in the anterior septum, Factor I, and a closed field in the accumbens nucleus, Factor II) which determine the forms of the AEPs in this region (Freeman & Patel, 1968).

The results show that _x(T, y, z) can be represented as the sum of a dipole field v_i(X), and a closed field v_ii(X). For each field the amplitude, ƒ_i(T) and f_ii(T), respectively, varies with poststimulus time, as shown by Fig. 4.19. Note that the movement of the zero isopotential surface of _x(T, y, z) shown in Fig. 4.17 is not evidence for movement of the emf in the field. According to the results the emfs of the two sets are fixed in location but variable in amplitude. They are asynchronous, although not statistically independent, in the sense that their product moment correlation coefficient is not zero. The asynchrony causes movement of the zero isopotential surface in the sum of the two component fields. §

4.3. Potential Fields in the Olfactory Bulb

Detailed analyses are given in these and the following sections of the field potentials of the olfactory bulb and prepyriform cortex. This is done in order to demonstrate some techniques for comparing observed and predicted fields, and to present some concepts and data, which are required in the next two chapters. The examples given for both bulb and cortex show the fields of AEPs. The bulbar and cortical EEG fields of potential are shown <Page 212> closely to resemble the evoked fields by comparing relative amplitudes of evoked and EEG activity recorded simultaneously at multiple points in the two neural masses.

4.3.1. BULBAR GEOMETRY AND TOPOLOGY

FIG. 4.21. View of the lateral aspect of the forebrain of the cat. Small square: 2 × 2 mm recording array (8 × 8 electrodes). Large square: 3.5 × 3.5 mm recording array (8 × 8 electrodes). Rows of white dots and black dots: 1 × 8 electrode arrays for stimulation, respectively, of PON and LOT. OM: olfactory mucosa; OB: olfactory bulb; AON: anterior olfactory nucleus; RF, rhinal fissure; PON: primary olfactory nerve; LOT: lateral olfactory tract; OT: olfactory tubercle; PC: prepyriform cortex; ES: entorhinal sulcus.

Three aspects of the anatomy of a neural mass must be known. These are the global geometry of the neural sets comprising the mass, the characteristic geometry of the main cell types, and the topology of the connections, which may be lines of flow of neural activity. These anatomical properties have been studied in detail by light and electron microscopy (Ramón y Cajal, 1955; Le Gros Clark, 1957; Valverde, 1965; Rall et al., 1966; Price & Powell, 1970a–d; Pinching & Powell, 1971a–d; Willey, 1973). Only the salient features are introduced here. In particular the rich variety of <Page 213> synapses found in the bulb by electron microscopy is not considered explicitly.

FIG. 4.22. Horizontal section through the forebrain of the cat at the level of the anterior commissure AC. Arrows show the direction of view of Figs. 4.21 (upper arrow) and 4.34 (lower arrow). PA: periamygdaloid cortex; PC; prepyriform cortex; V : ventricle. Nissl stain.

1. Global geometry. The gross anatomy of the olfactory bulb and cortex is shown in Fig. 4.21, which is a view of the lateral aspect of the brain of the cat, after the dura, skull, and soft tissues have been removed. The bulb has the size and shape of a lima bean. The flat medial and lateral (as shown) surfaces are parallel to the sagittal (midline) plane. The olfactory or <Page 214> prepyriform cortex has a curved surface that extends 2 cm posteriorly from the posterior edge of the bulb. The olfactory receptors lie in the olfactory mucosa that lines the nasal cavity and extends over a centimeter anteriorly from the bony plate at the anterior edge of the bulb.

The in–depth structure of the two masses is shown in a low–power view (10 ×) of a horizontal section (Fig. 4.22) in which the nuclei in the cell bodies have been stained (methylene blue). Each neural mass has a laminar structure in which groups of cell bodies form layers of grey matter parallel to the surface. The input axons to each mass form a layer at the surface, and the output axons contribute heavily to the mass of myelinated axons (white matter) at the inner aspect or base of each mass. The layers in the bulb are folded around a fluid–filled cavity called the ventricle. Domains limited to the lateral or medial walls can be treated as planes. The whole surface can be treated as a segment of an ellipsoid or, in the case of the bulb in the rabbit, as an incomplete sphere (Rall & Shepherd, 1968). The anterior half of the prepyriform cortex can be represented by a segment of a hemicylinder (Horowitz & Freeman, 1966).

FIG. 4.23. (a) Vertical sections through the lateral wall of the olfactory bulb that are perpendicular to the PON and to the bulbar surface.

FIG. 4.23. (b) Camera lucida drawings of neuron types in the olfactory bulb in the indicated layers. GL: glomerular layer; EPL: external plexiform layer; MCL: mitral cell layer; IPL, internal plexiform layer; GRL: granule cell layer; WM: white matter; P: periglomerular cell; T: tufted cell; M: mitral cell, G: granule cell; S: stellate cell; a: axon; x, o: basal dendrites extending into or out of plane of section. Rapid Golgi preparations (Freeman, 1972d).

The depth of the bulb is 1.6–1.8 mm from the surface to the ventricle. Three typical sections stained by different techniques are shown in Fig. 4.23a. Each shows the prominent masses of synaptic complexes called glomeruli that form a layer (GL) between the input axons in the primary olfactory nerve (PON) and a mass of interwoven dendrites and axons forming the <Page 215> external plexiform layer (EPL). A thin layer of cell bodies, the mitral cell layer (MCL), separates the EPL and the internal plexiform layer (IPL) that consists mainly of small axons. Beneath the IPL there are multiple laminae of cell bodies comprising the granule cell layer (GRL). At the base there is a layer of white matter (WM) that contains efferent axons and centrifugal axons (from other parts of the brain to the bulb). The efferent axons form the lateral olfactory tract (LOT) on the surface of the prepyriform cortex. These seven layers are all parallel to the surface (cf. Fig. 4.22).

2. Local geometry. The principal four types of neurons in the bulb are shown in camera lucida drawings from Golgi preparations in Fig. 4.23b. The finely branched terminals of each PON axon are restricted to a part of one glomerulus. Small neurons known as periglomerular neurons (P) occupy the glomerular layer (GL) and have axons and dendrites radiating in directions parallel to the surface. From their radial symmetry they are expected to generate closed fields.

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The spatial structure of the glomerular layer is shown in a low–power composite photomicrograph of a histological section cut parallel to the bulbar surface (Fig. 4.24a). The outside dimensions of the figure are 3.5 × 3.5 mm and are equal to the dimensions of the recording array shown by the larger rectangle on the bulb in Fig. 4.21. (The larger inset is 2 × 2 mm, the size of the smaller rectangle.) Each glomerulus is a mass of axonal and dendritic endings with small numbers of cell bodies. Large numbers of cell bodies are found between the glomeruli, which give the appearance in cross section of a mosaic of paving stones set in gravel. Each glomerulus is partially or wholly encapsulated in a layer of glial processes (larger inset, Rio Hortega stain). The cell bodies in the intervening spaces are mainly those of periglomerular cells. An example is shown from a Golgi preparation at the same size scale as the glomeruli (the marker is 100 µm). The dendrites <Page 217> extend between glomeruli for distances up to 4 or 5 glomerular diameters, but generally less far.

FIG. 4.24. (a) Composite photomicrograph of a section in the plane of the glomerular layer (GL in Fig. 4.23). Outer square is 3.5 × 3.5 mm; inner square is 2 × 2 mm (see squares in Fig. 4.21). The marker represents 100 µm for the scale of the two insets. Outer square: cresyl violet stain to show the nuclei of the periglomerular cells and the mosaic of the glomeruli. Inner square: Rio Hortego stain to show the partial glial encapsulation of the glomeruli. Small Inset: Golgi stain to show the size and structure of representative periglomerular cells.

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FIG. 4.24. (b) Reconstruction from serial sections of the outlines of glomeruli (cresyl violet stain) in a 1 × 1 mm section. View is from inside the bulb looking outwardly. The glomeruli are irregularly shaped globules, often confluent, with an average diameter of 135 µm.

A perspective drawing made by reconstruction of serial sections through the layer (1 × 1 mm, Fig. 4.24b) shows the irregular globular form of the glomeruli. Each has very roughly the shape of a sphere, as suggested by the cross sections in Fig. 4.24a, but they are both irregular and confluent. From measurements of the maximal diameters of the clumps in the reconstruction <Page 218> (as well as others not shown), the mean glomerular diameter is 135 ± 30 µm. The density is 18.3/mm², and the total is about 3200 in each bulb.

The next layer, called the EPL, is organized very differently. There are large numbers of long dendrites extending in all directions parallel to the surface at all depths. These arise from large neurons with cell bodies in the EPL called tufted cells (T) and in the MCL called mitral cells (M). Each has an apical dendrite normal to the surface extending into a part of one glomerulus or occasionally two glomeruli, and each has three or four basal dendrites extending for long distances (up to 1 mm) in all directions parallel to the surface. Each has an axon with multiple collaterals (side branches), which branch off as it extends to the white matter. From the high degree of radial symmetry the mitral and tufted cells are predicted to generate closed fields on generalized activation, or dipole fields when the active membrane is restricted to the glomerular tufts of the apical dendrites.

The many small neurons with cell bodies packed together in the GRL are called granule cells. Each has a few short basal dendrites and a single sparsely branched ascending dendrite oriented normal to the surface, which is studded with gemmules. The axial symmetry implies that these neurons generate dipole fields, when input is restricted to either end. The neurons have no axons, which implies that they do not generate action potentials.

Scattered through the EPL and GRL are relatively large neurons with dendrites and axons radiating in all directions. They are called stellate cells (S) and are predicted to generate closed fields when the input is distributed over the dendrites. Unlike the other three types, they do not densely populate a single layer.

3. Topology. The topology of connections defining the neural sets in the olfactory bulb (and cortex) is shown in Fig. 4.25. Each receptor (R) has one axon that extends to the bulb in the PON and ends in one of the glomeruli. The ending is a densely branched tuft (Fig. 4.23b). The approximately 140 million receptors for each bulb have no interconnections and form the KOR set. Periglomerular neurons (P) receive axodendritic synapses from PON axons, are densely interconnected by axodendritic synapses in and between glomeruli, and form the KI set. The mitral–tufted cells (M, T) receive axo– and dendrodendritic synapses from PON axons and periglomerular neurons. They form the KI_M set by virtue of their axodendritic and axosomatic synaptic interconnections through axons in the EPL and IPL. The granule cells (G) interact and form the KI_G set, though the channel of the interaction is not known. The mitral–tufted cells both deliver and receive synaptic input to and from granule cells, because most of the dendrodendritic synapses between mitral–tufted and granule cells are reciprocal (Shepherd, 1972). The interconnection between the mitral–tufted <Page 219> KI_M set and the granule cell KI_G set forms a KII_MG set in the neural mass of the olfactory bulb (see Fig. 1.5d).

FIG. 4.25. Schematic diagram of principal types of neurons, pathways, and synaptic connections in the olfactory mucosa, bulb, and cortex (see Fig. 1.5) (Freeman, 1972f).

4.3.2. ANALYSIS OF THE SPATIAL FUNCTION OF POTENTIAL

The coordinates of the lateral wall of the bulb are designated x (increasing in the main direction of orthodromic (forward) propagation of action potentials on the PON axons), y (in directions normal to the PON and parallel to the surface), and z (in directions normal to the surface and the PON). On single–shock stimulation of the PON at x = y = 0, the AEP, monopolarly recorded and averaged from a site on the activated bundle (x = x₀, y = y₀, and usually y₀ = 0) at the surface (z = 0) which is _x,y,z(T) shows a triphasic action potential followed by a complex response consisting <Page 220> of a damped sine wave superimposed on a nonoscillatory baseline shift (Fig. 4.26). The successive negative peaks (upward) are labeled N1, N2, ....  The successive positive peaks are labeled P1, P2,..., even if they are less than zero. The amplitude of the action potential attenuates rapidly to zero with increasing x (see Section 2.3.3).

The field of potential manifested by the AEP, (T, x, y, z), is broadly distributed over and through the lateral wall of the bulb. The field recorded with 64 electrodes (Fig. 4.27a) on the bulbar surface z = 0 is _T,z(X, y) and is shown for several representative values of T in Fig. 4.27b–g, including the rising phase, crest, and falling phase of N1, and the crests of N2, N3, and N4. The potential is unimodally distributed over a portion of the surface (the 8 × 8 rectangular array of electrodes is 3.5 × 3.5 mm), and the crest N1 moves across the surface in the direction of propagation of the PON axons. The crests of N2, N3, ... recur at increasing values for x.

The surface location, x = x₀, y = y₀ and z = 0, of the maximum for the potential of N1 over all values of T, x, and y, is called the epicenter of the bulbar response domain. The potential field in a plane transecting the bulb at x = x₀, _T,x(y, z), is shown in Fig. 4.28 for three values of T corresponding to the crest of N1, the zero crossing between N1 and P1, and the crest of P1 During N1 there is a dipole field with a zero isopotential surface in the deep half of the EPL, a negative pole in the EPL, and a positive pole in the GRL. During the zero crossing at the surface there is a closed field with a negative pole centered in the lower half of the EPL. During P1 there is a dipole field with a zero isopotential at the MCL, a positive pole in the EPL, and two negative poles in the IPL (see AEPs in Fig. 4.38).

FIG. 4.26. Typical bulbar AEP, following electrical stimulation of the PON with N = 335 (Freeman, 1972a).

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FIG. 4.27. (a) Set of 64 AEPs, _z(T, x, y), constructed from simultaneous recordings at the surface z = 0. Array size: 3.5 × 3.5 mm (see Fig. 4.21). Top: anterior. Left: dorsal. Amplitude: 200 µV. Time: 50 msec. (b)–(g) Maps of fields of potentials _T,z(x, y) at times of (b) onset, 11 msec, (c) crest, 16 msec, and (d) decline, 21 msec, of the first peak of the AEP, N1 (see Fig. 4.26), and at crests of (e) N2, 40 msec, (f) N3, 64 msec, and (g) N4, 86 msec (Freeman, I974a).

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FIG. 4.28. Maps of fields of potential _T,x(y, z) in a cross section of the bulb perpendicular to the PON and to the bulbar surface. Contour intervals are in microvolts. Poststimulus times refer to the crest of N1, the time midway between N1 and P1, and the crest of P1 (see Fig. 4.38) (Freeman, 1972d). (a) T = 13.1 msec at 60 µV; (b) T = 22.5 msec at 20 µV; (c) T = 29.4 msec at 20 µV.

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The center of the response domain is at x₀, y₀, z₀, where z₀ is the depth of the zero isopotential surface at N2, N3,  ....  The potential as a function of time at selected points along an electrode track through the epicenter is designated _xo,yo(T, z). The two examples in Fig. 4.29 are AEPs in response to PON stimulation (a) and LOT stimulation (b) for selected values of z. The AEPs have the same form but with different amplitudes, except near z = z₀. For z < z₀ and _xo,yo(T, z). the amplitudes of N1 and of the baseline shift are negative; for z > z₀, the amplitudes of N1 and of the baseline shift are positive. At and near z = z₀, the amplitudes of N1 and the baseline shift are minimal, and the oscillation lags about 90° (1.57 rad) from the oscillation recorded at the surface _xo,yo,z=0 (T).

FIG. 4.29. AEPs _xo,_vo(T, z) from an electrode track through the epicenter of a response domain on (a) PON and (b) LOT stimulation and recorded in the olfactory bulb of a cat (Freeman, 1972d).

For analysis by stimulation the following assumptions are made: (1) ρ(x, y, z) = ρ is constant. (2) The input velocity θ(x) is constant over x, and there is negligible delay in the y and z coordinates. (By this assumption we neglect events seen near z = z₀ during the rising phase and crest of N1, when the distribution of emf is being established in the EPL). (3) The number of sets is µ = 2. [By this assumption we omit consideration of action potentials in the PON and mitral–tufted cells, which are more suitably <Page 224> dealt with in the context of single axons and KO sets of neurons–see Section 2.33, 4.1.4, and 4.1.5, and reports by Rall (1962), Rall and Shepherd (1968), and Nicholson and Llinás (1971)]. (4) The representation of source sink geometry for each local subset of neurons by equivalent point charge is the same for every local subset within the same set. (5) The representation of the time dependence of charge magnitude for each local subset is by the <Page 225> same time function within the set. (6) The potential fields of all subsets are superimposed in the mass.

There is inferred to be a dipole field v_G(X) as in Fig. 4.11 which is generated by the KI set of granule cells. These are the only neurons having axial symmetry, which extend across the zero isopotential surface. There is inferred to be a closed field v_M(X), as in Fig. 4.12, generated by the mitral–tufted cells, which lies in the EPL, MCL, and IPL layers. These are the predominant radially symmetric neurons in or near those layers. The time function for f_G(T) is inferred to be represented by the AEP at the epicenter _xo,yo,zo(T). The time function for ƒ_M(T) is inferred to be represented by the AEP at the center 1000(T). The potential field of the response domain is inferred to be

FIG. 4.30. Depth gradient of potential _T,xo,yo,yo(z) at crest of N1 on PON stimulation. Dots: means ±SE (N = 4); solid curves: from Eqs. (69); dashed curves: from Eqs. (66) (Freeman, 1972d).

The parameters of the dipole field v_G(X), are evaluated in the following way. The values for potential along an electrode track through the epicenter v_T,xo,yo(z) are shown as means and standard errors for four sets of measurements in Fig. 4.30, for T al the crest of N1 on PON stimulation. At this time, ƒ_M(T) = 0. A predicted potential function is calculated as follows. The subset of granule cells in each local domain ∆x∆y is treated as equivalent <Page 226> to a core conductor, which is infinite in extent and uniformly depolarized over its superficial half z < z₀. From Example A in Section 4.1.3, the distributed source and sink are represented by an equivalent charge density distribution

where k = 1, ..., 50. The density of the active state of each subset and the current source–sink density is inferred to decrease monotonically in all directions of x and y from x₀ and y₀. The distribution is approximated by a bivariate normal density function that determines the values for the source density in each local subset over the whole set (Fig. 4.31a):

where σ_x, and σ_y are standard deviations, and i = 1, ..., 50, j = 1 . . . , 50. The charge at each point is

FIG. 4.31. (a) Calculated relation between the standard deviation a of a normally distributed set of activated granule cells and the half–amplitude radius of the surface field potential v_T,z(X, y) at z = 0 (Freeman, 1972d). Solid line: z₀ = 1000 µm; dashed line: z₀ = 800 µm.

For calculation of v_T,xo,yo(z), the delay in activation T is fixed at one value T_a for all q_a because the contributions from q_a for all x < x₀, where <Page 227> T < T_a, are added to those from q_a for x > x₀ where T > T_a. The potential along the z axis at x₀ – y₀ = 0 is

Curves for v_T,_xo,_yo(z_n) are fitted to _T,xo,yo(z) by trial and error selection of values for λ and z₀ (Fig. 4.30). The optimal values are λ = 170 µm, z₀ = 820 µm, and σ_x = σ_y = 900 µm.

FIG. 4.31. (b, c) Measurements of surface distributions of potential _T,xo,z(y) and _T,yo,z(X) from the bulb of a cat on PON stimulation at three times during Nl. Fitted curves: from Eq. (69) with λ = 170 µm and z₀ = 820 µm. Inset shows locations of 1 × 8 recording arrays and the half–amplitude ellipse (Freeman, 1974a). (b) N = 100 at 1.5 × threshold for 3 pps. O: T = 14.4 msec, σ = 560 µm; •: t =17.6 msec, σ = 465 om; ∆ : t = 20.8 msec, σ = 320 µm. (c) O: t = 14.4 msec, σ = 770 pm; •: t = 17.6 msec, σ = 880 µm; : ∆ : t = 20.8 msec, σ = 880 µm.

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The mean length of 82 granule cells in the cat is 320 µm (Freeman, 1972d), so the electrotonic length is 320/170 or 1.88. This is similar to the value 1.7 estimated for the mean electrotonic length of the granule cells in the rabbit (Rall & Shepherd, 1968). The summation in Eq. (69) is truncated at z – z₀ = ±3λ, or 510 µm. This corresponds approximately to the range of distribution of the dendrites of granule cells over the combined depth of the EPL, IPL, and GRL. The range is greater than the mean length, because the cell bodies are distributed through the GRL.

 The potential computed at the surface z = 0 is sufficiently far from the dipole source–sink distribution that q(z_k) can be replaced for each subset by a point source and sink q(z_k) = 1 at z_k = z₀ + l and q(z_k) = –1 at z_k = z₀ – l. For fixed values of z₀, l, σ_x, and σ_y, the potential at the surface along any line through the epicenter is given by

and Eqs. (67) and (68). The predicted surface potential is a unimodal bivariate distribution in x and y (Fig. 4.31a).

The half–amplitude distance x_h or y_h of the response domain is the distance from the epicenter of activity to a location at which the surface amplitude is half the amplitude at the epicenter. The value for y_h increases with increasing σ_y at fixed values for l and z₀. The relation is linear (Fig. 4.31a).

Also,

Equation (71) holds to a good approximation for σ > 300 µm. Below that value for σ_y, y_h approaches a minimal value of 600 to 700 µm, depending on z₀. The value for σ_y is insensitive to changes in σ_x or l. Equation (71) suffices to predict the standard deviations of a bivariate random distribution of active granule cells, for which the mean is beneath the epicenter, from the half–amplitude widths of the surface field of potential (Fig. 4.31b and c).

4.3.3. TIME–DEPENDENT ACTIVITY

The bulbar response is established over a domain in the bulb by propagating action potentials in the PON at a mean velocity θ_x = .42 m/sec. The questions are considered whether the delay in the PON can account for the apparent movement of the field of potential across the bulb, whether the effect of the delay persists as a phase gradient across the response <Page 229> domain, and whether a phase gradient exists radiating from the epicenter, similar to the phase gradient of waves radiating from the site of an object dropped in water. Answers are sought by measuring the frequency and phase of the oscillatory component of the AEPs at each of 64 points, and by predicting the phase gradient using the distributed dipole model.

The delay is introduced into the array of fixed modules representing source–sink density in the response domain as an ordered delay in onset of activity T_x, increasing in x by T_x = x_i/θ_x where θ_x = .42 m/sec is PON mean conduction velocity (Section 2.3.3). Once activity is established in each local subset of granule cells, the activity is assumed to have the same time function, but with varying amplitude dependency on location. The function is estimated from the AAEP, an ensemble average, v_x(T) = [_{x,y,z = 0} (T)], of the set of 64 AEPs recorded at the surface.

The AAEP is measured by fitting it with the sum of basis functions (Section 2.5.3) specified by v_x(T) = v₁(T) + v₂(T):

where v₁(T) represents the damped sine wave oscillation and v₂(T) represents a baseline shift (Section 2.5.3). The frequency of the damped sine wave is ω in radians per second; the phase is φ in radians; the decay rate is in reciprocal seconds; the rise time of N1 is ß in reciprocal seconds. The rise time, decay rate, and rate of recovery from a positive overshoot of the baseline shift are, respectively, b, c, and a.

An example of the AAEP is shown in Fig. 4.32a by the plotting symbols; v_x(T) is shown by the solid curve, and v₁(T) and v₂(T) are shown by the dashed curves. The relation is

where (T) is the minimized least squares deviation.

Equation (72) is then fitted to each of the 64 AEPs from the surface electrode array. Parts (b) and (c) of Fig. 4.32 show the isopotentials based on the 64 values for V₁ and for V₂ (see Fig. 4.27c). The frequency ω varies insignificantly over x and y. The 64 values for decay rate α and phase φ are represented by contours in, respectively, parts (f) and (e) of Fig. 4.32. The arrow denotes the line y = z = 0. The observed values for φ(x) at y = 0, z = 0 are plotted in the part (d) as the filled circles. The straight line segment indicates the nature of a uniform phase gradient.

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FIG. 4.32. (a) Average AEP (∆). Solid curve: fitted from Eq. (72); dashed curves: dominant and minor components (see Section 2.5.3). (b) Contour plot for amplitude V₁ in Eq. (72). (c) Contour plot for amplitude V₂. (d) Comparison between observed phase (•) and predicted phase (∆) from Eq. (72) fitted to the wave forms constructed with Eq. (74) with θ = .42 m/sec mean PON conduction velocity. (e) Contour plot for phase φ. (f) Contour plot for decay rate α.

The potential as a function of time for selected points along the line x_n for fixed y = 0 and z = 0 is

where y is summed over half the field by virtue of symmetry. The function v_y,z(x_n,T) is the predicted waveform for the AEP at each site x_n. Comparison with the observed AEPs is made most easily by fitting v_y,z(x_n, T) with Eq. (72). The predicted frequency ω(x_n) for the damped sine wave is everywhere the same. The predicted amplitude distributions for V₁ and V₂ over x are very similar to the observed distributions for σ_x = 900 µm and σ_y = 720 µm. The predicted phase φ(x_n, y, z) at y = z = 0 (triangles) is compared with the observed phase φ(x, y = 0, z = 0) in Fig. 4.32d. The direction and amount of change in phase lag with increasing x_n, as predicted <Page 231> from the designated values of θ, z₀, λ, σ_x, and σ_y introduced into the model, are close to the observed change over x_n = x_i.

The predicted and observed AEPs differ in one important respect. Whereas the predicted value for α(x_i), the decay rate of the damped sine wave in Eq. (72), is everywhere the same, the observed values are distributed. High values are found at the epicenter and low values at the margin of the response domain. This is not compatible with the assumption of constancy of v_G(T) over the response domain. This is an important property that is discussed in detail in Section 6.1.4.

Analysis of responses to low–level LOT stimulation shows that the same general properties hold, but the conduction velocity is much faster, and the direction of spread is reversed, as expected from the direction of antidromic propagation (in reverse of the normal direction of flow) of LOT input to the bulb (see arrows in inset in Fig. 4.31b, c).

Analysis has also been carried out for single–shock responses to supramaximal stimulation of the LOT (Rall & Shepherd, 1968). In this case the delay in activation is assumed to be fixed, and the distribution of input is assumed to be to the entire set of mitral cells in the bulb. The gross geometry of the sets of mitral and granule cells in the rabbit is represented by a "punctured sphere." The predicted and observed absolute amplitudes v_t,x,y(z) and v'_t,x,y(z) are greater inside the bulb z < z₀ than at the surface z = 0 for every value of t.

The phase gradient in depth σ_xo,yo(z) is also useful. A set of AEPs on low–level PON stimulation as a function of depth _xo,yo(z, T) is fitted with the basis functions designated in Eq. (72). Additionally a set of AEPs on low–level LOT stimulation is fitted with another set of basis functions (Section 3.5.3)

The values for φ and φ_m are plotted as functions of depth in Fig. 4.33. The phase is constant with increasing depth to within about 100 µm of a reversal point, and then, within an additional 200 µm of depth, reverses by an average of 3.13 rad (3.14 rad = 180°). The narrow range of phase reversal z₀ ± 100 µm is evidence that the contribution of the closed field in the EPL, MCL, and IPL is much less in amplitude and spatial extent than the contribution of the dipole field.

There are two spatial components in both the orthodromic and antidromic fields of potential evoked by PON and LOT electrical stimulation. The dominant component v₁(T, X) is a distributed dipole field V_G(X) with its zero isopotential surface in or near the MCL, which is assigned to the granule cells. One pole is in the EPL and the other pole is in the GRL.

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FIG. 4.33. Phase of oscillatory components of AEPs as a function of depth in bulb along a line passing through the epicenter of response domains (Freeman, 1972d).

The secondary component v_ii(T, X) is a closed field v_M(X) with its main pole located in the EPL and the MCL, and which is attributed to the mitral–tufted cells. The time function of the dipole field ƒ_G(T) has an oscillatory transient superimposed on a baseline shift. The time function of the closed field ƒ_M(T) is oscillatory. The oscillation lags the oscillation in ƒ_G(T) by about one–quarter cycle. Explanation of the significance of the time functions is deferred to Section 5.4.4.

Next we consider the relation between v_xo,yo,z(T) and the active state of the KI_G set. The activation of the set depends on input paths with constant conduction velocities. There is a delay in onset of activation of each local subset, which is a function of conduction distance. The oscillating impulse response manifested in the dipole field at the surface _x,y,z=0(T) displays a phase gradient in the direction of propagation of the input. Because the delay is continuously increasing along x across the epicenter at x = x₀ and because the activity on both sides of the epicenter is in the form of a damped sine wave, the sum of waves from subsets for x < x₀ with phase lead and of waves from subsets for x > x₀ with phase lag is equal to the phase of activity of subsets for x = x₀, in the formation of an ensemble average over subsets near the epicenter. Then we need not consider the delay in further analysis of the active state in the neighborhood of the epicenter.

At and near the center of the dipole field the zero isopotential surface is <Page 233> plane and the isopotential surfaces are parallel to it. The lines of extracellular current of the field are normal to the isopotentials and parallel to the axes of the generating neurons oriented perpendicular to the surface. The set of conditions is formally identical to the condition of a nerve suspended in air in which the extracellular action current is constrained to flow parallel to the axons and in the reverse direction of intracellular current. The extracellular potential function at any time T with depth from the epicenter, T, _T,xo,yo(z) is proportional to the intracellular potential function along the axes of the neurons, except that the scale and reference potential for the two fields may differ (see Section 4.1.3, Example D).

From this principle we infer that an extracellular AEP recorded on a line passing through the center and epicenter is proportional to the time–varying average transmembrane potential over an ensemble of subsets of neurons surrounding the center of the response domain, which contribute current to the line at the center. This geometric condition holds if the site of recording at z lies outside the domain of the closed field centered at z = z₀. The average deviation in transmembrane potential difference from resting potential v_m(T) – v₀ is defined as the active state in the wave mode for a subset of neurons (Section 1.3.2). Then the AEP from any point between the epicenter and the superficial pole of the dipole field is a measure of the active state of the subset of neurons at the center of the response domain.

where v_m – v₀ is the average deviation from rest of the transmembrane potential of each KI_G subset near the site of transmembrane current reversal over z; k is an empirical proportionality constant; and _G[^•] denotes the ensemble average over KI_G subsets in the vicinity of the center of the response domain of the olfactory bulb.

Equation (76) holds for other recording sites located on the line joining the epicenter to the superficial pole, because the phase gradient along that line segment is zero (Fig. 4.33). Because of the smoothing property of summation of potential, it holds to a good approximation within the half amplitude width of a response domain. For a plane dipole field it holds for sites in and deep to the deep pole z > z₀, provided the sign of k is reversed. For curved dipole fields that are convex at the surface, it holds for both poles. The domain of the ensemble average [v_m(T)], which is related to by Eq. (76), is asymmetric with respect to sites on the outer side z < z₀ and inner side z > z₀. The ensemble domain is smaller on the outer side and larger on the inner side. In summary, the proportionality expressed by Eq. (76) between the AEP and the KI_G active state depends on the local and global architecture of bulbar neurons and on their time and space activity distributions. Within the specified limitations, Eq. (76) provides <Page 234> the principal basis for testing the dynamic models described in Chapters 5 and 6.

4.4. Potential Fields in the Prepyriform Cortex

4.4.1. CORTICAL GEOMETRY AND TOPOLOGY

FIG. 4.34. Lateral view of forebrain of a cat at the angle shown by the lower arrow in Fig. 4.22. Rectangle is 4.0 × 7.2 mm (6 × 10 electrode recording array). • : represent 1 × 8 electrode array for stimulation. DL: dorsolateral bulb; VL: ventrolateral bulb; DM, DL, VL, VM: locations of axons in the LOT from the designated quadrants of the bulb; Al, A2: parts of anterior olfactory nucleus AON; PC: prepyriform cortex; RF: rhinal fissure; ES: endorhinal sulcus; OT: olfactory tubercle; MCA: middle cerebral artery; V, W, X, Y: selected recording sites (see Fig. 4.38).

The primary olfactory cortex is coextensive with the area of termination of the LOT axons on the ventrolateral surface of the brain. It is divided into two main parts, frontal and temporal, where it is crossed by the middle cerebral artery (MCA, Fig. 4.34). Only the frontal part is shown. This part is further divided into the anterior olfactory nucleus (AON), which includes the areas in Fig. 4.34 marked Al and A2, and prepyriform cortex <Page 235> (PC). The LOT covers most of the lateral surface of the nucleus in its trajectory from the bulb to the other parts of the primary olfactory cortex. It converges to the medial edge of the prepyriform cortex. Its axons and axon collaterals turn laterally, diverge, and intersperse over the surface of the cortex. The distribution is diffuse over the entire cortex extending from the entorhinal sulcus (ES) medially to the deepest part of the rhinal fissure (RF) laterally.

The lateral aspect of the nucleus (AON) in the cat is sufficiently fiat to be represented by a plane. The axons in the LOT in its trajectory over the nucleus are topographically arranged in such a way that the most dorsal axons come from the dorsomedial quadrant of the bulb (DM), and the most ventral come from the ventromedial quadrant (VM). Those between come from the dorsolateral (DL) and ventrolateral (VL) quadrants. The bulb projects into the LOT as if it were cut on its medial surface and spread fiat (Shepherd and Haberly, 1970). Mitral–tufted axons from the medial bulb divide to pass above and below the ventricle in the bulb (Fig. 4.22). This arrangement implies that there is a degree of topographical order in the bulbar projection to the nucleus, such that neurons in each horizontal strip of the bulb project to a horizontal strip in the nucleus.

The cortex (PC) can be represented geometrically by a hemicylinder of radius r. The long axis of the cylinder is parallel to the LOT and is labeled x. The y axis is along the diameter of the cylinder, and the z axis is perpendicular to the midpoint of the cylinder (Fig. 4.35). The LOT axons converge to the ventromedial edge of the hemicylinder, and their branches turn 90° to diverge over the convexity of the hemicylinder. There is no detectable topographic order in the distribution. Axons from each part of the bulb end in all parts of the cortex.

FIG. 4.35. Schematic diagram of the geometrical pattern of spread of activation over the hemicylindrical prepyriform cortex after single–shock stimulation of the LOT (from Horowitz & Freeman, 1966).

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One effect of the curvature is that much of the nuclear and cortical surface is buried in the rhinal fissure (Fig. 4.34) forming the lateral border. Plane arrays of electrodes are placed only over the exposed part of the surface by applying sufficient light pressure to the surface (Fig. 4.34 in which the 4 × 7 mm rectangle represents a 6 × 10 electrode array).

The thickness of the cortex is 1.0–1.2 mm. There are three main layers (Fig. 4.36). Layer I is called the molecular layer and is divided into la, the LOT, and lb, a layer of superficial dendrites and axon collaterals. The outer half of Ib contains LOT collaterals, and the inner half contains axon collaterals of cortical neurons. Layer II is formed by densely packed cell bodies of small neurons called superficial pyramidal cells. Layer III contains the cell bodies of deep pyramidal cells, short axon cells, and polymorphic cells. The cortex is bounded at its base by a mass of myelinated axons that is part of the white matter of the interior of the brain.

FIG. 4.36. Histological sections through the prepyriform cortex, showing the layers with three stains: (a) Nissl; (b) Bodian; and (c) Weil. The arrows mark the inner border of Layer IA containing the axons from the bulb (Price, 1973).

There are three main neuron types (O'Leary, 1937; Ramón y Cajal, 1955; Valverde, 1965; Heimer, 1968; Stevens, 1969; Price, 1973; for nomenclature, see the work of Pigache, 1970). The superficial pyramidal or polymorphic cells (Type A) have multiple short, branched, apical dendrites (Fig. 4.37a) extending from the cell bodies in Layer II mainly into Layer Ib. Their axons descend and radiate into Layers Ib, II, and III. They receive axodendritic synapses from LOT axon collaterals over the superficial half <Page 237> and axon collaterals from other cortical neurons in the deep half of their extent in Layer Ib. Because of the axial alignment of their dendrites and the location of synapses in Layer Ib, they are expected each to generate a dipole field. On activation only by LOT axons, the field is expected to be asymmetric with the greater source–sink density and absolute amplitude of potential in the superficial pole. This is because the active membrane areas are limited to the superficial terminals of the dendrites (see Fig. 4.2a and b). On activation, however, also by axons of intracortical origin, the dipole field is expected to be less asymmetric due to the extension of the active membrane to the area of the soma. In the second case, the zero isopotential surface should be located nearer the surface (see Fig. 4.42 in the next section) relative to the zero isopotential surface in the first case (see also Fig. 5.31).

FIG. 4.37. (a) Distribution and appearance of prepyriform superficial and deep pyramidal–type cells drawn from Golgi preparations of immature mice. (b) Distribution and appearance of four short–axis cylinder cells (prepyriform interneurons or granule cells) with axons extending principally into Layers Ib and II (from O'Leary, 1937, with omission of his distinction of layer III into two parts III and IV).

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The second type of neuron is characterized by dendrites radiating in all directions for relatively short distances from a cell body in Layer III, but not into the outer third of Layer Ib (Fig. 4.37b). Its axon ascends and radiates into Layers Ib, II, and III. This type has been labeled as "deep neuron," "pyramidal cell," "granule cell," and "short–axis cylinder cell" by various authors. It is here called the "Type B cell." The star–shaped radial symmetry of its dendrites gives the expectation of a closed field of potential on uniform synaptic activation.

The third type of neuron, called deep pyramidal cell, has a pyramid shaped or irregularly shaped cell body in Layer III and relatively extensive branched dendrites in Layers II and III (lowest cell, Fig. 4.37a). The axon of this cell is known to give off collaterals in Layer III and occasionally Layer II, and its main axon is known often to leave the cortex (Valverde, 1965). The cortex is known to have a well–developed efferent axon system (Heimer, 1969; Price, 1973), and in keeping with the cytoarchitecture of other areas of cortex, it is reasonable to believe that the larger neurons maintain the axons forming the efferent system. The efferent neurons of the cortex comprising the third type are grouped here under the label "Type C" neurons.

The proposed principal connections are shown schematically in Fig. 4.25. The Type A neurons receive input from the LOT and are densely interconnected with each other by axon collaterals in Layers Ib and II. They are known to be mutually excitatory (Section 1.3.1), so they form the KI_A set. The Type B neurons receive input from Type A neurons but not from the LOT, and they are densely interconnected with each other. They are mutually inhibitory and form the KI_B set (Section 5.4.4). Their axons have dense connections onto the dendrites of Type A neurons, which is the basis for the existence of a KII set in the cortex. The intracortical connections of the efferent neurons, here called Type C are inadequately known and they are provisionally designated as the KO_C set. By virtue of the laminar geometry, the KI_A set of Type A neurons is expected to generate an asymmetric distributed dipole field centered in Layer lb, and the sets of Type B and C neurons are expected to generate closed or nearly closed fields centered in Layers II and III.

4.4.2. OBSERVED FIELDS OF CORTICAL POTENTIAL

Single–shock electrical stimulation of the LOT at the surface in the midregions of the cortex, z = 0, results in a compound action potential in the LOT. This propagates orthodromically over the cortex along the x axis at a velocity of 5 to 10 m/sec. The amplitude decreases rapidly with increasing x (Fig. 4.38b). The volley is carried over the convexity by collaterals <Page 239> at a lower velocity (1–2 m/sec). It is followed by an oscillatory evoked potential, which is shown after averaging as a selected set of AEPs, _x,y,z(T) in Fig. 4.38b. The positions of the monopolar recording and stimulating electrodes are shown in Fig. 4.34 by the black rectangle and the letters, Y, X, W, and V, and by the 1 × 8 array of black dots.

The frequency of the oscillation differs for different locations on the cortical surface. In this respect the cortical AEPs are unlike the bulbar AEPs on PON stimulation which have a common frequency (Fig. 4.38a).

FIG. 4.38. Comparison of AEPs _x,y,z(T) from (a) surface of bulb on orthodromic PON stimulation and from (b) surface of cortex on LOT stimulation. Recording sites on cortex are shown in Fig. 4.34. Dashed lines: preservation of a phase gradient along the cortex due to conduction delay in the LOT; amplitude: 200 µV; time: 20 msec.

The potential at the surface v_T,z(x, y), z = 0, is shown in Fig. 4.39 for selected instants during the first surface negative peak, N1. Each arrow has its tail at the location of the maximal potential in that frame and its head at the location of the maximum in the succeeding frame. There are two maxima of potential in the surface coordinates. The anterior maximum overlies the anterior olfactory nucleus (A2). It shows little tendency to move during N1 and recurs at approximately the same location during N2, N3, etc. The posterior maximum moves upwardly in the frame (dorsolaterally over the cortex, PC) in the direction of the LOT axon collaterals. By the conclusion of N1 in the anterior part of the cortex, the peak is moving into the inaccessible part of the lateral surface.

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FIG. 4.39. Surface distributions of potential _T,zo(x, y) at selected times during Nl of the cortical AEP on LOT stimulation (sec Fig. 4.34). Time: (a) 1.28 msec; (b) 384 msec; (c) 5.12 msec; (d) 6.40 msec.

FIG. 4.40. Surface distributions of potential _T,zo(x, y) at selected times indicated by letters, A, B, C, D, in Fig. 4.38 and corresponding to parts (a), (b), (c), (d), respectively.

The distribution of potential over the surface during subsequent peaks, P1, N2, P2 is rather complicated. Four representative distributions are shown in Fig. 4.40. The patterns suggest that the impulse input initiates <Page 241> oscillatory events in two domains of the cortex underlying the electrode array. The potential fields of the domains partially overlap, so that the locations of the epicenters cannot be precisely determined by inspection. The successive maxima in potential over the nucleus (Al) occur in the same location, whereas the successive maxima of potential oscillation over the cortex move dorsolaterally.

FIG. 4.41. Depth distribution of single–shock response v'_T,m,x(y z) at the time of the maximal amplitude of the first surface–negative, deep–positive peak. Use of this technique gives rise to overlapping isopotentials in the lateral aspects of the field due to conduction delay. (a) A12; (b) A14; (c) A16; (d) A18 in millimeters anterior to midaural plane.

The depth distribution of potential of single–shock response v'_T,m,x(y, z), is shown for four coronal planes in Fig. 4.41. The planes of mapping at A18 and A16 shown by parts (d) and (c), respectively, pass through the frontal part, and those at A14 and A12, parts (b) and (a), respectively, pass through the temporal part of the cortex. The planes lie about 30° from the yz plane that is perpendicular to the LOT. The values at T_m are taken from the peak of N1 recorded at each site without regard to its <Page 242> latency, in order to represent the maximal amplitude distribution in one set of figures. As the result, the isopotentials for the peak positive and negative values cross each other over the lateral convexity, where the delay is greatest. The field is a curved distributed dipole field. During N1 the superficial pole is negative and is centered near the LOT. The deep pole is positive and is located within the concavity of the curvature. The zero isopotential surface is located near the base of Layer Ib, where the trunks of the dendrites approach the cell bodies of the superficial pyramidal cells.

The precise location of the zero isopotential surface for any point in depth is determined by making a small lesion or electrolytic deposit with a microelectrode. When the location of selected points has been histologically verified, the change in location of the zero isopotential surface with time is determined with respect to the verified points. Partly because of the technical difficulties in making and reading a large number of small lesions, and partly because of spatial distortions in processing the histological sections of the brain in which the recordings are made, the maps of isopotentials based on selected sampling sites are subject to uncontrollable local distortions. The sections and the maps are only approximately superimposable.

An example of the movement of the zero isopotential surface _x(T, y, z) = 0, which implies the movement of all other isopotential surfaces, is shown in Fig. 4.42 at intervals of 1.25 msecs from the crest of N1. By the crest of N1 it is established as a curved surface AA, part (a) corresponding to the curvature of Layer II and known from use of electrolytic deposits to lie in the inner third of Layer Ib. During the next 3.75 msec (to DD) it rotates around the convexity of the cortex. By 5 msec from the crest (EE), part (b), the polarity of the dipole field is totally reversed. The zero isopotential surface is reestablished in Layer Ib, but during P1 it continues to rotate about the convexity, similarly to a rotating tangent to the base of Layer Ib.

The pattern of spread is shown in the schematic diagram in Fig. 4.43a of a cross section of the cortex through PC in Fig. 4.34 (see also Fig. 4.37). On the right is shown a selected set of recording sites [A–F, part (b)] and a representative set of AEPs from those sites [part (c)]. Each element represents a subset of cortical neurons that generates a dipole field. The start latency increases with distance from the medial edge of the cortex. One effect of the delay is seen at the start of the AEP labeled F, where the initial deflection is downward (positive). This is due to recording the deep positivity of the dipole field established at A and B before the activating volley has reached E and F. The same phenomenon can be visualized in maps of the contours of potential superimposed on photomicrographs of the plane of mapping. In this case the isopotentials move with poststimulus time.

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FIG. 4.42. Location of the zero isopotential surface of the field of potential _T,x(y z) in the plane perpendicular to the LOT and to the cortical surface. (a) Locations during N1; (b) locations during P1.

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FIG. 4.43. (a) Sequence of activation of prepyriform cortical dipole generators (see PC in Fig. 4.34) following LOT stimulation. (b) Location of elements deep to surface recording sites. (c) AEPs at designated sites in the plane A17 (with permission of J. M. Horowitz).

The dipole field is attributed mainly to the KI_A set of Type A neurons, because these neurons have the appropriate local and global geometry and the required location of synaptic input to account for the position of the zero isopotential surface and the locations of both poles. This conclusion holds for both surface–negative and surface–positive polarities of the dipole field. It is supported by intracellular recordings from Type A cells, which show EPSPs followed by IPSPs (LOT stimulation, Biedenbach & Stevens, 1969b); by the effects of small lesions in the cortex, which reduce both N1 and P1 if made in Layer Ib and have no effect if made in Layers III and IV (Fig. 5.32); and by source–sink analysis based on differencing (Haberly & Shepherd, 1973; see footnote in Section 4.1.2).

In summary, both the anterior olfactory nucleus and the prepyriform cortex generate dipole fields of potential. Both oscillate at some characteristic frequency on impulse driving, and both oscillations are initially surface negative. Both dipole fields reflect an oscillatory active state in the KI_A sets in the respective parts of the primary olfactory cortex. The location of the oscillatory active state in the nucleus bears some topographic relation to the part of the LOT which activates it, and the location of the event is fixed. The oscillatory active state in the prepyriform cortex shows a systematic <Page 245> lag in the direction of delay in initial activation. That is, there is a phase lag or phase gradient in the direction of LOT conduction delay. It is the phase gradient that gives the appearance of a rotating dipole field. There is no present evidence for the existence of a topographic relation between the bulb and the prepyriform cortex.

4.4.3. RELATION OF POTENTIAL FIELDS TO ACTIVE STATES

The chief desideratum in extracellular recording in multineuronal fields of potential is a measure of potential  that is proportional to the ensemble means of the active states of the generating neurons (v_m). We have seen (Section 4.3.3) from the analysis of a plane dipole field that if a coherent domain can be identified, the potential at the epicenter is a function [Eq. (76)] of the active state of the neural subset comprising the domain. This principle also holds for epicenters on the convexity of curved surfaces. It cannot, however, hold if the surface potential is determined by dipole fields of two or more KI sets forming a neural mass. Types B and C neurons are expected to generate closed fields on the basis of their local dendritic and synaptic geometry, but the possibility exists that incomplete radial symmetry may lead to the formation of a dipole component of the fields of subsets of Types B and C neurons that could degrade the relation between the surface potential and the active states of the set of Type A neurons.

The problem is analyzed in the following way. The cortical mass consists of three sets of neurons. The KI_A and KI_B sets are more densely populated than the KO_C (or possibly KI_C) set, so only these two are considered. The KI_A set generates a dipole field v_A(X) with its zero isopotential surface at the base of Layer I. The active state for impulse input can be assigned the time function of a damped sine wave ƒ_A(T) = V_A sin(ωT + φ_A)e^–α^T. The KI_B set generates a field of potential v_B(X), which is inferred to be a closed field centered in Layer II. The active state also has the time function of a damped sine wave ƒ_B(T) = V_B sin(ωT + φ_B)eα^T. The frequency and decay rate in f_A(T) and f_B(T) are identical, but the active state of KI_B must lag by one–quarter cycle the active state of KI_A. (This will be proven in Chapter 5.) Therefore, φ_B – φ_A = –90°.

The field potential at the surface v_x,y,z(T), z = 0, reflects only the closed field v_B(X) and the time function f_B(T). The field potential in Layer II, v_x,y,z(T), z > z₀, is the sum of the deep pole of the dipole field and the monopole field v_A(X)+v_B(X). The time function is the vector sum of ƒ_A(T) and ƒ_B(T), which is v_ABsin(ωT + φ_AB)e^–α^T, where φ_AB depends on the phases and relative amplitudes of the two damped cosines.

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These phase relations are shown in Fig. 4.44. In parts (a)–(c) there are, respectively, two AEPs from the deep pole that are initially positive (downward) in potential and a third AEP (the lowest) that is from the surface pole and is initially negative (upward). The active state of the KI_A set is shown by the lower PSTH, which oscillates in phase with the surface AEP. The active state of the KI_B set is shown by the upper PSTH, which oscillates with a quarter cycle phase lag from the surface AEP. (These phase relations also hold between the pulse probability waves of Types A and B units and the cortical EEG as shown in Sections 3.3.3, 3.3.5, and 5.4.4). This implies that the surface AEP has the same phase, frequency and decay rate as the active state of the KI_A set in this stimulus–recording arrangement.

FIG. 4.44. Evidence that a prepyriform field of evoked potential consists in the superposition of a dipole field generated by the Type A neurons v_A(X) and a monopole field generated by the Type B neurons v_B(X). The time function ƒ_A(T) is obtained in the AEP recorded at the cortical surface [(d), lower AEP], and the time function f_B(T) is obtained in the AEP recorded at the zero isopotential surface of v_A(X), where v_B(X) is near maximal [(d), middle AEP]. Elsewhere the AEP reflects the weighted sum of ƒ_A(T) and ƒ_B(T) [(d), upper trace], as shown in (e) the vector diagram. The forms for ƒ_A(T) and ƒ_B(T) are predicted from PSTHs as in (a)–(c) (see also Section 5.4.4) (Freeman, 1968b). (a) N = 450 for 450 µm at 125 msec; (b) 270 µm; (c) –300 µm ; (d) N = 92 for the three curves at 370, 0, and –180 µm, respectively, at 62.5 msec. Distances are heights above zero isopotential in µm.

In part (b) the lower AEP is from the surface recording and conforms to a damped cosine wave, cos(ωT)e^–α^T. The phase φ_A = 90°, and relative amplitude v_A are shown by the vector at A in part (e) at 90°. The middle AEP is from the base of Layer I at the minimum for the amplitude of peak N1. The amplitude v_AB is relatively low, and the phase φ_AB is 180°. The AEP in fact conforms to an initially positive damped sine wave. The KI_B set is maximally excited at the first downward crest of the sine wave. The Type B neurons in the set have dense current sources in their cell <Page 247> bodies and broader but less dense current sinks in their radial dendrites that are concentric withtheir cell bodies. As shown in Figs. 4.10 and 4.12, the high density of the source gives the closed field its polarity. The initially positive (downward) sine wave, therefore, represents a field of potential v'_B sin(ωT + 180°)e^–α^T, which is 180° out of phase with the active state ƒ_B(T) = v_B sin(ωT + 0°)e^–α^T, That is, the active state ƒ_B(T) lags the active state ƒ_A(T) by 90°, but the field potential v’_B leads the active state ƒ_A(T) by 90°. This is shown by the vector B in part (e). The upper AEP in part (d) is taken from Layer II. The potential predicted for the deep pole of the dipole field is v’_Asin(ωT + 90°)e^–α^T, because it is 180° out of phase with the surface recording. This is shown by the vector –A. The vector sum of –A and B predicts the phase and amplitude of the AEP from the deep pole, which typically has about 30° of phase lag from the inverted surface AEP. It is concluded that the cortical field consists of a KI_A dipole field and a KI_B closed field, and that surface recordings reflect the active state of the KI_A set.

Another determinant of the phase of AEPs in the cortex is the combination of curvature and delay in activation (see also Section 4.3.3). This combination, however, gives a different pattern of phase relations than that attributable to the KI_A and KI_B sets. This is shown by setting up a hemicylindrical array of dipole charge elements (Fig. 4.35), and assigning to each element a delay in impulse input, and an impulse response consisting of a damped sine wave. The potential is calculated for selected points representing specified recording sites. The order and amount of delay for each element, the size and curvature of the cylinder, the frequency and decay rate of the impulse response, and the distances of the recording sites from the zero charge surface at z₀ are adapted to correspond to the dimensions of the cortex and its response.

Predicted responses at three points on the surface are shown in Fig. 4.45a. They show the predicted phase lag of the oscillation in the direction of LOT impulse propagation. The predicted functions of potential with depth are

(Fig. 4.45a). There is a disparity between the absolute amplitudes at the inner and outer recording sites. For the designated parameters of frequency and delay, the outer absolute amplitude exceeds the inner absolute amplitude |v₁(T)| > |v₂(T)|. The relation is the reverse of the intuitively predicted relation (Example B in Section 4.2.2 and Section 4.3.3). It occurs because the potential at site 2 inside the cylinder is the sum of contributions that <Page 248> are weighted more heavily toward those from the limits of the cylinder than is the case for the site 1 outside the cylinder. This results in amplitude reduction inside the cylinder owing to phase dispersion. It occurs in the prepyriform dipole field (Fig. 4.41), as shown by the greater absolute amplitude of the surface pole over the deep pole.

FIG. 4.45. (a) Predicted phase lag of calculated AEPs that is designed to conform to experimental observations (Fig. 4.43). Dashed line: V₁; dotted line: V₂; solid line: V₃. (b) Calculation of the predicted AEPs at the cortical surface (1), midpoint (2), and depth (3) shows that the absolute amplitude outside the cylinder V₁ exceeds the absolute amplitude inside the cylinder V₃. This conforms to observation (Fig. 4.41) in reverse of the expectation for a curved dipole surface without afferent delay. However, the predicted AEP at the center of the dipole sheet V₂ does not conform to observation (middle AEP, Fig. 4.44d). Therefore, the cortical field of potential is not accounted for by a curved dipole sheet generator with activation delay; a monopole generator must be included (from Horowitz & Freeman, 1966).

Although the weighted summation across the response domain is sufficient <Page 249> to reverse the ratio of absolute amplitudes, it is not sufficient to change the phase relation between ₁(T) and ₂(T) by the amount required. Moreover, there is no significant phase shift in the predicted function ₃(T) with respect to the mirror image of the function ₁(T).

It is concluded that the surface potential _x(T), x = x₀, y = y₀, z = 0, is proportional to the ensemble average of the active states of the subset of neurons in the KI_A set [v_m(T)] which forms a coherent domain centered under the recording site. Therefore Eq. (76) derived for the KI_G set in the olfactory bulb holds also (within the specified limitations) for the KI_A set. The same conclusion cannot be drawn for surface recording sites at the margins of coherent domains or for sites beneath the cortical surface or the zero isopotential surface.

4.5. Divergence and Convergence in Neural Masses

4.5.1. THE OPERATION OF DIVERGENCE

The active state o_µ of each set µ in a neural mass can be described by the functions of time, amplitude, and distance o_µ(t, x, y, u) , where t is real time, X is x, y, and z, and u is a variable representing amplitude in either pulse mode p or wave mode v. Under the assumptions given previously, each function and mode can be separated into three parts:

The global potential field generated by the mass and manifesting its active states in the wave mode v'(t, X) is the sum of the fields of potential of a number of sets M that make detectable extracellular contributions in the wave mode. The procedure has been described for postulating on anatomical grounds the existence of a number of sets in a mass, determining the probable location and spread of active states by locating afferent tracts and synaptic sites, predicting source–sink distributions from local and global neural geometry, and calculating a potential function v_µ(X) for each set. This function is weighted by a characteristic time function v_µ(X) ƒ_µ(T) for each set, and the sum over M is evaluated from (T, X).

Each function ƒ_µ(T) is dependent on the active state in the wave mode o_µv(t, x, y, u) . Each function v_µ(X) is determined by q_µ(X), representing the source–sink distribution. The time dependency of the active state ƒ_µ(T) for a local subset of neurons is given by the ensemble average of transmembrane potential for the neurons in the subset (v_m(T)]. Certain conditions have been described in which this is proportional to _x(T). We now ask how the <Page 250> equivalent charge function q_µ(X) depends on the active state h_µv(x, y). Two transformations are needed:

The first transformation H_1v is the step of converting a proposed activity density function o_µm(t, x, y) to the accompanying source–sink distribution and then to a distribution of equivalent charge. An example has been given in Section 4.4.3 for the KI_B set. The neurons are radially symmetric and are concentrated in Layer II of the cortex. When they are excited, they generate concentric source–sink distributions with central high–density sources. In large numbers the fields of individual neurons add without vector components, so the expected field of the excited KI_B set is a closed centrally positive field. This is represented by a concentric charge distribution q_B(X) with its outer limits in the deeper part of Layer T. The second transformation H_2v is the step of summing the potential contributed by all the charge to determine the potential field v_B(X).

The observable electrical activity of a neural mass manifests distributions of activity in the pulse mode o_µp(t, x, y, u) . The pulse trains recorded and averaged in a mass, p'(t, X) and (T, X), are generated by a number of sets M. The averaged pulse density function of each set p_µ(T, X) is separated into time– and distance–dependent parts p_X(T) and p_T(X), where p_µX(T) may be determined by PSTHs, and p_T(X) is found by recording pulse trains at multiple sites in an active domain. Again, two transformations are defined and evaluated:

where ψ_µ(X) is the spatial (three–dimensional) distribution of impulses determined by the active state q_µp(x, y) and is homologous with h_µ(X), and where p_µ(X) is the distribution of action potentials determined by electrical recording. ^† Again, the first transformation H_1p is the prediction of the relation between an active state and a pulse density function. The predicted pulses may occur in the dendrites, in axonal initial segments, or for some neurons <Page 251> not at all. The second transformation H_2p is required to establish the spatial relation between the spatial distribution of pulses and a spatial field of potential. The field may be quite different from the distribution of pulses, as for example the compound action potential is not the same as any single action potential (Fig. 2.11) and in fact may be everywhere zero in the presence of a large volley of action potentials (Fig. 2.14). In general, ψ_µ(X) is not identical to p_µ(X) because the field of the action potential extends beyond the neuron generating the impulse.

^† An assumption that has not been thoroughly tested but can and should be with the multiple microelectrode technique described in Example A in Section 7.2.2 is that the pulse density function p_µT(X) is constant over z through the depth of the KI_µ set as well as over small distances x and v. This assumption is needed in order to establish an equivalence between ψ_µ(X) and h_µp(x, y) by the operation H_1p in Eq. (81). Some evidence for its validity is found in the fact that the neurons in each column in somatosensory and primary visual neocortex have very similar receptor field properties (Hubel & Wiesel, 1962; Mountcastle. 1966).

For any set that contributes both to (T, X) and (T, X), if the active states of the set are defined with respect to transmembrane potential and pulse probability at or near the trigger zone of each neuron, then

More generally, the conversions from waves to pulses and from pulses to waves that are the bases for defining G(v) and G(p) in neural masses (Section 3.3.6) can be conceived as taking place at points, so that it is feasible to define a single function h_µ(X, y) for each neural set in a mass, which is independent of mode and may be estimated from observed activity in either mode.

Evaluation of the transformation implied by H_2v has been undertaken in the preceding sections. Analysis and evaluation of H_1v, H_1p, and H_2p is given in the following sections. Transmission of activity from one set µ to another µ + 1 involves two complementary processes. The projection of many neurons in set µ onto each neuron in set µ + 1 is convergence. The projection of each neuron in set µ onto many neurons in set µ + 1 is divergence. If either occurs, both must occur, and the choice of label is determined by whether the input or the output is identified on a single neuron. If divergence occurs, then the spatial functions of the active states for the sets are related but not equal. Let h_µ(x, y) denote h_µv or h_µp depending on the mode for the µth transmitting subset. Then the relation

defines the operation H of divergence of neural activity on transmission from the active state of set µ to the active state of set µ + 1.

The spatial distribution of an active state is defined over the xy plane of a neural set. Divergence takes place in the xy plane in directions orthogonal to the topological trajectory of transmission. The description and measurement of divergence is based on measuring two appropriate functions, _µ(x, y) and _µ+1(x, y) in the x and y dimensions with appropriate sets of basis functions in x, and on comparing the parameters of the sets of basis functions. The most useful basis function is the normal density function. The measure of the active state on this basis in the x and y coordinates is <Page 252> the variance σ_µ² (the square of the standard deviation σ_µ) and σ²_µ+1 of the normal curves fitted to them. The operation of divergence is expressed as the ratio σ²_µ+1/σ²_µ, or the difference σ²_µ+1 – σ²_µ of the output and input variances.

Divergence takes place during transmission at synapses and over compound nerves and tracts. The mechanisms differ, so we will distinguish between two basic classes of divergence–convergence and call them, respectively, synaptic divergence and tractile divergence.

Tractile divergence has three mechanisms. Each tract or nerve has a set of cross–sectional areas over its trajectory. The tract is coextensive with the areas of the transmitting and receiving neural sets at the start and end of each tract, but many tracts are funneled through small areas in transit, by dense packing of the axons. If a volley is induced in a small area of high density by an electrical stimulus, the set of action potentials spreads out over the target neural set. The dilation of the volley from onset to termination is in proportion to the ratio of output to input cross–sectional areas. The number of pulses is not changed. This is dilative divergence. If the axons of neurons in each local subset of a transmitting set become interspersed among the axons of other local subsets on arrival at the receiving set, the representation of the transmitting subset (after any correction for dilation) is larger at reception than on transmission. If there is no branching, neither the total number of pulses nor the pulse density is affected. This we call interspersive divergence. If each axon gives off collaterals that are interspersed with the collaterals of other axons, both the total number of pulses and the pulse density are increased. This we call collateral divergence. Dilative divergence is described by the ratio of variances. The other forms of synaptic and tractile divergence are described by the difference between variances.

The complementary facet of divergence H(X) is convergence H^–1(X), in which each point on the receiving surface receives from a distribution of points on the transmitting surface. In this point of view the various types and mechanisms of divergence do not appear explicitly. The operation is most familiar in the form of the receptor field of a single sensory neuron (e.g., Barlow, 1953; Hubel & Wiesel, 1962; Mountcastle, 1966) from which input converges to that neuron. That is, the input to the neural system is a spatial distribution of excitation and/or inhibition, and the output is the pulse rate of a single neuron or a unit cluster. The convergence operation H^–1(X) can also be described by means of Gaussian basis functions (Rodieck & Stone, 1965), and where both H(X) and H^–1(X) have been measured in the same preparation (see Fig. 4.46) the two facets of the operation are found to be congruent. Typically, only one of the two facets of operation is accessible to measurement, but there is no definite reason at present to question the complementarity, so that either type of measurement should suffice.

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4.5.2. EVALUATION OF SPATIAL DISTRIBUTIONS OF ACTIVE STATES

The spatial distribution of an active state h_µ(x, y) is defined by an activity density function (see Section 1.3.2) at any time T over the surface normal to the direction of transmission z. The current source–sink distribution of the active state is represented by an array of fixed charge q(X) that has been tested by comparing its potential field v(X) with an observed field T(X). In order to determine the activity density function, an operation must be performed on the array of charge to determine its extent in the x and y dimensions. The nature of the required operation depends on the system under study and its mode of operation.

^a Sequence in list shows order of transmission through olfactory bulb.

^b Observable experimentally.

^c Dorsal: σ_gd, σ_a•gd, etc.; Ventral: σ_gv, σ_a•gd, etc.

Some appropriate examples are taken from the olfactory system of the cat. The main sequence of events following monopolar single–shock electrical stimulation of the PON is listed in Table 4.2 (Fig. 4.25) The stimulus current excites a certain distribution of axons in the PON that conducts action potentials to the glomeruli in the bulb. There is dilative and interspersive divergence between the stimulus site and the glomeruli. The PON input excites distributions of periglomerular neurons in the glomerular layer and mitral–tufted cells in the EPL and MCL. The mitral–tufted cells excite a <Page 254> distribution of granule cells that then inhibits another distribution of mitral–tufted cells. The widths of distributions have been measured for PON axons, periglomerular neurons, excited mitral–tufted cells, excited granule cells, and inhibited mitral–tufted cells.

Example A. The compound action potential of the PON is shown in Fig. 2.13. The set of traces on the left is from monopolar recording at surface sites along the z axis of transmission, x₀ = 0, y₀ = 0, _xo,yo(T, z). The set of traces on the right is from surface sites, x₀ = 0, along a line y normal to the transmission axis _xo,zo(T, y) at a value for z₀ of 1.0 mm from the stimulating electrode at z = 0. The x axis is normal to the surface of the PON. ^†

A time T₀ is selected at which the value _To,xo,yo(z) = v₀(0), maximally negative, corresponding to the negative–upward crest of the compound action potential. The functions, _To,xo,yo(z) and _To,xo,zo(y), are digitized. The values are normalized by dividing each by the maximum v₀(0). On the basis of the analysis given in Section 4.1.3, the function _n,To,xo,yo(z) is treated as proportional to the source–sink density along z and is used to specify a source–sink distribution along the z axis:

The potential as a function of distance from the transmission axis along y is calculated at n points

The values are normalized by division by v₀(0). The half–amplitude width of the potential function is less than that observed, so Eq. (86) is not valid.

^† The convention being followed here is that the z axis conforms to the main direction of transmission in a neural set, and the xy plane is the set of directions in which divergence takes place. When divergence is evaluated in the PON, the z axis lies parallel to the PON axons. When divergence is evaluated in the bulb, the z axis lies perpendicular to the PON on the bulbar surface, and the x axis is aligned with the PON. We use a convention based on local functional topologies rather than on anatomical or stereotaxic coordinates for the whole brain, because the tracts and layers in the brain are usually curved and are not readily described in Cartesian coordinates. Hence, each computation of divergence must be preceded by specification of the coordinates.

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Next, the active axons are postulated to lie in a bivariate random distribution h(x, y) around the z axis with the mean at x₀ = y₀ = 0 and with standard deviation σ_a. The distribution of charge in x and y is

where h(x, y) is represented by a bivariate normal density function. The potential along y is

The values of v(y) are normalized by division by v₀(0). The value of σ_a is varied, until the observed and computed half–amplitude widths x_a of the fields of potential coincide. The empirical relation of x_a to σ_a is approximately a straight line for σ_a > 50 µm:

in microns. An example of the observed potential (points), calculated potential (curve), and equivalent charge distribution in y (dashed curve) is shown in Fig. 4.46. The average of 31 measurements is σ_a = 190±49 µm

FIG. 4.46. Experimental (filled and open points, ±SE, N = 12) and predicted distributions of potential [solid curve; Eqs. (85)–(90)] as a function of the perpendicular distance (y) from the axis (x) of a set of activated axons, which are distributed about the x axis in accordance with the normal density (dashed curve, σ_a = 300 µm). See Fig. 2.13b (Freeman, 1974c).

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(SD). This is the desired mean value and range for the activity density function of the PON on monopolar stimulation h(x, y) when measured with the normal density curve as the basis function.

The PON axons are parallel to each other over their main trajectory so that the same results are obtained whether the stimulating electrode [to determine H^–1(X)] or recording electrode [to determine H(X)] is moved in steps over y (Fig. 4.46, respectively, filled and open points). §

Example B. The evaluation of the activity density function for periglomerular neurons on monopolar PON stimulation is based on measurements of pulse rates of single neurons in PST histograms (Section 1.3.3), _xo,yo,zo(T), where T is poststimulus time. The x axis is parallel to the direction of PON transmission and passes through the epicenter of the response domain at x = x₀, as determined from AEPs (Section 4.3.2); y is the line normal to x at the surface through the epicenter at y = y₀ ; z_p. is the depth of the glomerular layer. The mean pulse rate is found for a time interval ∆T starting at the onset of an evoked response of the neuron and ending 10 msec later (Fig. 5.10).

The monopolar stimulus site is changed to successive sites along y as the basis for determining H^–1(X), and the PSTH from a pulse train at a fixed recording site is obtained on stimulation at each stimulus site. The distribution of the mean induced firing rate for 17 neurons is shown in Fig. 4.47 over the range of variation in stimulus site. The epicenters have been superimposed. The values are fitted with the normal density function with _p = 410 µm. On the basis of results shown in Fig. 4.46, it is inferred that the function relating mean response rate at one site on variation of input site is congruent with the variation in mean response rate over a set of recording sites for a fixed input site.

The field of the action potential around each neuron may be detected as far as 100 µm from the maximal potential at or near the neuron. If 100 µm is adopted as an outside value for the standard deviation σ_vp of the distribution of the field of the action potential, the observed variance _p² is the sum of variance due to the distribution of the active state σ_p² and the variance σ²_vp due to the field of each neuron. For _p = 410 and σ_vp = 100, σ_p = 397. The difference between _p and σ_p is less than the error of measurement, so the inclusion of the effect of the field of the action potential is unnecessary, and σ_p = _p. That is, P_p(X) ψ_p(X) in Eq. (82) and h_p(X, y) is evaluated by _p. §

Example C. The distribution of potential at the surface as measured during N1 of the AEPs is inferred (Section 4.3.2) to be generated by a random distribution of active granule cells at a depth z₀ = 820 µm with the <Page 257> mean at the epicenter and standard deviations σ_x and σ_y in microns. The potential functions v_TN1,yo,zo(x) and T_N1,xo,zo(y) are fitted with curves from Eq. (70) to evaluate σ_x and σ_y. For the example in Fig. 4.47, y_h = 970 µm and σ_y = 560 µm. It is inferred that the source–sink distribution in x and y is identical to the activity density function, so that σ_g = σ_y.

FIG. 4.47. (a) Experimental (points ±SE, N = 17) and theoretical distribution (curve) of AEP amplitude during N1 at surface of bulb (see Fig. 4.31); x_h = 975 µm. (b) Experimental distribution (points, ±SE, N = 17) of periglomerular KI_P activity (σ_P = 410 µm) as a function of distance from the epicenter of response domains, compared with estimated distribution (wider curve) of KI_G activity (σ_g = 560 µm). Spontaneous rate is .07 spikes per trial lasting 10 msec (equivalent to 7 pulses/sec) (Freeman, 1974d).