Comput Biol.Med.
Pergamon Press1975. Vol.5.pp.X3-2%. Pnnted inGreatBritain
SIGNAL
DISPERSION WITHIN A HIPPOCAMPAL NEURAL NETWORK* J. M. HOROWITZ and J. W. B. MATES
Department of Animal Physiology, University of California, Davis, CA95616, U.S.A. (Received 18 December 1974 and in reaisedforrn 4 March 1975) Abstract-A model network is described, representing two neural populations coupled so that one population is inhibited by activity it excites in the other. Parameters and operations within the model represent EPSPs, IPSPs, neural thresholds, conduction delays, background activity and spatial and temporal dispersion of signals passing from one population to the other. Simulations of single-shock and pulse-train driving of the network are presented for various parameter values. Neuronal events from 100 to 300 msec following stimulation are given special consideration in model calculations. Neural model
Hippocampus
tl rhythm
1. INTRODUCTION Neighboring pyramidal neurons of the hippocampus are oriented in parallel and lie in a single sheet. Macroelectrodes, placed in this sheet of cells, sometimes record a slow (4-7 Hz in the rabbit), high amplitude EEG wave called the “theta rhythm.” The hippocampus and its 8 rhythm are of special interest because they have been related to arousal, learning and memoryll-21. Models of hippocampal neural networks have been put forward in attempts to shed light on hippocampal function and malfunction [3-61. A neural model was previously developedE61 representing a recurrent inhibitory feedback network of hippocampal pyramidal cells and interneurons [7-91. It allowed the simulation of experiments and the comparison of calculated and experimental hippocampal behaviors. Using arguments like those of Freeman[ lo] regarding neural populations with inhibitory feedback and spontaneous activity, damped oscillatory evoked potentials (AE Ps) and poststimulus time (PST) histograms had been expected and were recorded from appropriately stimulated cat hippocampus[ll]. Using a minimum number of parameters [ 121, the hippocampal model’s behavior was consistent with experimentally measured hippocampal responses to single-shock excitation of pyramidal cells. Furthermore, model simulations successfully predicted the hippocampal response to pairedshock stimulation. Factors modifying simulated hippocampal waveforms were studied and were presented together with the results of related experiments[6]. The new hippocampal model described here relaxes an assumption which, although convenient, was not biologically tenable; namely, that signal delays between pyramidal cells and interneurons were all identical. The long-term dispersing effects of relaxing this assumption were unclear. By placement of a convolution in the present model’s feedback branch, these dispersive effects were simulated and examined. Further waveforms were calc:ulated for pulse-train driving and for long-term behavior (i.e. over many hundreds of * This project was supported in part by NASA Grant NGR-05-004-099. by University of California Grant D-529, and by Public Health Service Grant NIMH06686. 383
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milliseconds), inviting comparison of model outputs with hippocampal EEG. Specifically, this report describes the behavior of a model hippocampal network, (1) modified to include dispersive effects, (2) driven by pulse-trains, and (3) observed over long time spans. 2. ANATOMICAL AND PHYSIOLOGICAL BACKGROUND 2.1 Anatomical connections of pyramidal ceils The model neural network represents two populations of hippocampal cells, the pyramidal cells, in a layer below the ventricular surface of the hippocampus, and the basket cells, lying adjacent to the pyramidal cells but closer to the ventricular surface (Fig. 1). Fibers terminating near pyramidal cells have been traced back to other pyramidal cells via Schaffer axon collaterals and the longitudinal association system, to the entorhinal cortex, to the septum via the fornix and to basket cells which synapse directly on the somata of the pyramidal cells in dense basket-like assemblages [ 13-151. Pyramidal cell axons are sent into the fornix, and collaterals of some of these axons synapse with basket cells and form the just-mentioned Schaffer collaterals and longitudinal association system. 2.2 Pyramidal cell electrophysiology
Despite the complexity of connections involving pyramidal cells, relatively simple functional networks have been identified by stimulation of the fornix which both orthodromically and antidromically excites pyramidal cells. Following such stimulation, Spencer and Kandel[l6] recorded long-lasting and powerful IPSPs in individual pyramidal cells. By sectioning the fornix and thus abolishing orthodromic input, they were able to show, by stimulation on the hippocampal side of the section, that the pyramidal cells could be antidromically activated and that the IPSPs were still observed. They inferred the presence of a recurrent inhibitory loop back to pyramidal cells. Andersen et al. [8] demonstrated that the IPSPs were produced at or near the pyramidal cell somata, and suggested that an interneuron was in the recurrent loop and that the interneuron was a basket cell [9]. The somatic location of the synaptic inhibition appears
Fig. 1. Schematic of pyramidal cell-basket cell neural network. Recurrent inhibition of pyramidal cells. P2, via their axon collaterals and basket cells, BA, is indicated by black arrows 1-3, 5-7. Arrow 1 represents input to pyramidal cells, Pl. arriving on fibers from the fornix. Arrow 4, on P l’s axon. represents pyramidal output traveling along the hippocampal surface to the fornix. Arrow 8 represents recurrent excitation of pyramidal cells, P 1, by other pyramidal axon collaterals, P 2.
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placed to maximize the modulatory effect of negative feedback on pyramidal cell output. Additional records obtained from antidromically activated pyramidal cells in such deafferented preparations were recently interpreted as evidence for recurrent excitatory activation of pyramidal cells [17]. Anatomical grounds for such recurrent excitation are available in the Schaffer collaterals and longitudinal association system, and again in the basket cells. Lebovitz[17] has pointed out, however, that the massive recurrent inhibition of pyramidal cells effectively obscures other weaker influences on pyramidal cell output. Whereas most intracellular records of pyramidal cells taken by Andersen et al. [8,9] showed prolonged poststimulus IPSPs, few cells showed short-latency (i.e. less than 5 msec lag) EPSPs or antidromically evoked spike potentials. Signals on axon collaterals of a few pyramidal cells may be sufficient to cause inhibition of many pyramidal cells. Durations of EPSPs and IPSPs were reported to average less than 10 msec and about 300 msec respectively. Even in the absence of applied electrical stimulation of the fornix, EPSPs and IPSPs are assumed to be constantly generated in the network as reflected by the steady-state rates of pyramidal cell spike generation [6,11,18]. And because of the wide dispersion of inhibitory feedback among pyramidal cells, one may expect both pyramidal and basket cell populations to have signficant mean firing activity levels.
ideally
2.3 Hippocampal EEG and behavior Pyramidal cell activity has been linked to the generation of the hippocampal 0 rhythm. For example, hippocampal pyramidal cell spiking activity is usually phaselocked to the 13 waves[20]. However, the drive for the rhythm has been traced to structures outside the hippocampus. Petsche et al. [21] showed that a class of cells in the medial septum fired rhythmic bursts of spikes in step with the hippocampal 0 rhythm; that during post-seizure hippocampal silence, these cells were silent, and that prior to every recurrence of the 13waves, the septal cells were again rhythmically firing bursts of spikes. Cross-correlations measured between septal and hippocampal EEGs and between septal unit activity and hippocampal EEGs show that the activity of certain septal cells (the “B” cells) closely correlates with the amplitude of the 8 rhythm, and that hippocampal EEGs lagged those of the septum[22]. Occurrence of the 8 rhythm in hippocampal EEGs has been correlated with alertingtl], with memory storage[23] and with a variety of other behaviorally-related events. The 8 rhythm can be induced by novel stimuli such as would tend to arouse an animal (e.g. light flashes, handclaps); cutaneous temperature transients were also found to evoke 0 activity in the unanesthetized, loosely restrained rabbit [24], adding yet another sensory modality to those previously shown to elicit this rhythm. 3. HIPPOCAMPAL MODEL HM2 The model HM2 is an extension of one described earlier 161,and is diagrammed in Fig. 2. (The properties of the pyramidal and interneuron populations selected for modeling are overlaid on outlines of a pyramidal and basket cell.) The signals passing through an element in the network represent averages over all the cells and fibers corresponding to that element. Provided one assumes the average poststimulus activity of any single neuron is the same as that of all other neurons of its type, one can relate experimental
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PYRAMIDAL
CELL
Fig. 2. The neural population model in block form. Pyramidal cell components include two EPSP elements, an IPSP element, and a nonlinear element T. A DISPERSE and DELAY element lie on the feedback branch which includes the basket cells. N represents steady background EPSP activity from INPUT,,. After a delay, a second element simulates EPSPs elicited by shock (impulse) stimulations delivered to input fibers at INPUT,. If the summed EPSPs and IPSP exceed a threshold, the non-linear T element delivers an output. The delayed and dispersed output over the feedback branch inhibits pyramidal cells.
data in the form of PST histograms of single pyramidal cells to calculated model outputs[6]. A special advantage of this formulation is that it allows the use of continuous signals in the model simulating the network. In Fig. 2, three inputs are shown summed by the modeled pyramidal cell population: (1) EPSPs following excitation of a fiber tract (at INPUT,); (2) EPSPs from a steady background excitation N (from INPUT,); and (3) IPSPs recurrently fed back onto the pyramidal population by interneurons presumed to be basket cells. Shock stimulation of fibers at INPUT, is assumed to cause a rectangular pulse (PULSE element of Fig. 2) of excitatory input to the pyramidal cell population after a delay (DELAY element at left of Fig. 2). This input causes depolarizations or EPSPs of the pyramidal cell membranes. If the sum of the EPSPs and IPSPs exceeds a fixed threshold, the pyramidal population produces a non-zero output level (T element, Fig. 2); otherwise the output is zero. The output excites interneurons (basket cells) which, after a delay, (right-most DELAY element, Fig. 2), inhibits the pyramidal cells by causing membrane hyperpolarizations or IPSPs. The DISPERSE element of Fig. 2 represents the lumped effect of individual cell parameter variations causing dispersion of otherwise synchronous activity around the feedback loop. This element convolves its input with a dispersing function to produce its output. Figure 3 is a flow diagram of the computation sequence corresponding to the blocks of Fig. 2. Background pyramidal excitation, N, is held constant during each simulation run. The stimulus-induced EPSPs and IPSPs are characterized: EPSP = x,
(1)
IPSP = -n * y, d.3 + k,i + c,x =
excitatory
(2)
input signal,
(3)
287
Dispersion in a hippocampal network
Fig. 3. Formulation of model HM2 for CSMP programming. The set of boxes with open circles in their upper left corners are for calculating poststimulus EPSP transients; boxes with filled circles are for calculating IPSP transients. *d&per (convolution) requires a Fortran subroutine as does the nonlinear T (threshold) element.
d,j + k,j + c,y = inhibitory input signal.
(4)
The constants c,, c,, d,, d,, k,, k,, are chosen so that the EPSPs and IPSPs induced by pulse inputs are overdamped transients which resemble experimentally recorded intracellular potentials [8,9] (In addition, the equations can be related to state space variables for transmitter release)[6]. The number of pyramidal cells excited by a single basket cell is proportional to n, a parameter in the feedback branch and in equation (2) for IPSPs. The non-linear element, T, behaves according to: OUTPUT=D-T where
for
DzT;
OUTPUT = 0.0
for
Dee T;
(5)
D = EPSP + IPSP + N.
T represents a critical population firing threshold analogous to that of an individual ceil. Temporal dispersion of the signals in the feedback loop is calculated as the following convolution by the DISPERSE element (Fig. 2) whose input and output at time, t, are FL(t) and F,+,(t), respectively where disper(t) is set to zero for t 50. FL+,(t) =
I
+= am
-r
*disper (t - t’) - dt’
compactly, F L+l = FL*disper.
(6)
The integral is approximated stepwise by means of serial products[l9]. For the simulations of this report, the “disper” function was rectangular with width, W. The total loop delay equals W/2 units plus the pure loop delay (DELAY element), so variations in
C.E
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W will cause variations in the total loop delay. Without compensating normalization of the “disper” function’s area, changes of W cause changes in the loop gain. The DELAY elements of Figs. 2 and 3 represent pure time delays. Multiplicative elements of Fig. 3 amplify signal levels by a fornix gain, gi, and a loop gain, gl. The PULSE element of Fig. 3 produces a rectangular output pulse in response to an input impluse. Equations (l-6), together with the flow diagram in Fig. 3, describe model HM2. The major assumption in the model is a particular connection of cell populations based on experimental data outlined in sections 2.1 and 2.2. 4. RESULTS The model is shown in Fig. 3 as formulated for the Continuous Systems Modeling Program (CSMP) described in IBM’s users manual H20-0367-1. Constants and inputs used in the simulations (shown in Fig. 3) described in this report are presented in Table 1. Figures 4-7 show simulated pyramidal output traces which are assumed comparable with experimentally acquired PST histograms accumulated from poststimulus spiking records of individual pyramidal cells. For all simulated output traces shown in this report (refer to the labeled trace A of Fig. 4), the first input signal (the leading edge of a pulse) arrived at the EPSP element immediately prior to the rise of the first peak, P,, of pyramidal output. A depression of pyramidal activity, D, (extending below the mean firing level by a displacement, &), was usually followed by a peak of pyramidal activity, Pz (extending above the previous resting level of pyramidal output by P,“). Each trace Table 1. Constants and parameters used in CSMP calculations of waveforms INPuTa:
Pulse height = 1.0; pulses Of Figure Input gain, Of Figure Farnix
for trace
A
= 0.520.
Background activity, N = 0.004.
INPuTb :
EPSP and
(except gi=2000).
gi=looo 4 where
delay
(except =O.OlO).
“idal=0.015 7 where width
IPSP:
X8
=
xb
=
dx = d
Cx kx
FEEDBACK LOOP:
Y = -700; =
y,
=
yb
=
0.0.
1.0. c
-
-200.
-loooooY k = -10000 Y
Threshold Loop gain, Dispersion
of s1
T element =
10;
= 0.0. lcmp delay
= 0.025.
fmmction, ‘disper,’ is rectangular viei width = 25,
heigh - 0.04. Fan O"t, " = moo.
Dispersion in a hippocampal network
Fig. 4. Calculated pyramidal waveforms as parameter N, representing background activity, was decreased (top to bottom). For N = 0 (trace E) no oscillations occur. With decreasing N the second peak, Pz, moves slightly away from the first, P,; but the later interval durations do not vary. Dots are placed 5 msec apart.
Fig. 5. Calculated pyramidal waveforms as parameter PF, representing delay in the feedback loop, was decreased (top to bottom). The arrows indicate the second poststimulus pyramidal activity peaks. Oscillation periods decrease with decreasing delay. Very short delays cause rapid damping of the waveform (bottom two traces).
289
290
J. M.
HOROWITZ and J.
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A
W. B.
5
S
p3
MATES
w
=I00
w=40
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5
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CA d
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4 \
p2
p3
w=2
Fig. 6. Calculated pyramidal waveforms as parameter W, representing the width of the normalized rectangular dispersing function in the feedback loop, was decreased (top to bottom). The waveform period decreases with decreasing W (because of decreasing total loop delay).
shown in the present figures represents 500 msec of continuous activity; the points on each trace are 5 msec apart and computer calculations were made in steps corresponding to 1 msec intervals. Finally, all traces shown start after initial transient activity of the network had declined to a steady level. Simulated pyramidal outputs following a single stimulus pulse are traced in Figs. 4-6 as various model parameters are altered. Figure 4 shows the effects of variations of the background activity, N, upon the pyramidal output. Variation of N has no effect on the long term oscillation frequency; although, with decreasing N, the time between first and second output peaks (P, and P,) increases slightly. Increasing N causes increased oscillation amplitude. With no background activity (N = 0), no oscillations occur. Figure 5 shows the effects of variation in the feedback loop delay. The longer the delay, the longer the period of the oscillatory output. Very short delays cause damping of pyramidal output. Feedback loop dispersion, W, is varied in Fig. 6. The heights of the rectangular “disper” functions were altered to compensate for changes of their widths (i.e. the rectangular dispersing function with which the feedback signal is convolved was normalized). Increasing W increases the total loop delay which causes the oscillation period to increase. Figure 7 shows simulated pyramidal output initiated by pulse-train driving at four different frequencies. At the lowest driving frequency (trace A), the network can pass each pulse to the output: but at the highest driving frequency (trace D), the network responded only to the initial step-like transient of mean excitation (by an oscillatory rise
Dispersion in a hippocampal network
291
Fig. 7. Calculated pulse-train driven pyramidal waveforms as functions of increasing driving frequency (top to bottom). For each trace, the first pulse of the train arrived at the EPSP element just prior to the first pyramidal activity peak. The impulse frequency at INPUT, (Figs2,3) was varied. Top to bottom, the impulse frequencies were IOHz, 50 Hz, 200 Hz and 1000 Hz, respectively. In response to each impulse, however, the PULSE element (Figs 2,3) generated a pulse of width 10 msec. Thus pulses arrived at the EPSP element at 10 Hz, 50 Hz, 66 Hz and 99 Hz. Pyramidal cell output level increases as the mean input increases (10 msec pulses at 66 Hz and at 99 Hz, lower two traces); note the poor high frequency response.
of pyramidal output to a higher steady level), and was unable to pass the input frequency to its output. Thus, the neural model behaves as a low pass filter, blocking high input frequencies and passing low input frequencies. 5. DISCUSSION 5.1 Oscillatory activity PST histograms for hippocampal pyramidal cells commonly display a succession of peaks[ll]. Prior to the shock excitation, pyramidal cells are modeled as firing at a constant rate. Therefore the simulated network is in an active state (except when background activity N is set to zero [Fig. 41). There are two available means of increasing simulated pyramidal output: (1) activity of input excitatory fibers can drive pyramidal firing frequency higher; (2) by contrast, suppression of inhibitory feedback allows normal excitatory activity to drive the firing frequency upwards. This latter case has been oblserved in the spinal cord and called “disinhibition” by Wilson and Burgess [25]. Peaks in model pyramidal cell activity, corresponding to peaks observed in oscillatory PST histograms, can thus be due either to added excitatory inputs or to disinhibition. Figure 8 illustrates how oscillatory network activity may develop following single pulse excitation of pyramidal cells (P). Parts a, b and c of the figure represent three states of the model network at successive times 1,2 and 3 in a continuum of states initiated by
J. M. HOROWITZand J. W. B. MATES
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I
2
3
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13
Fig. 8. Schematic “snapshots” of pyramidal inputs and outputs at three successive points in time. Parts a, band c represent times T = 1, T = 2 and T = 3, in the poststimulus history of the network. On the left are diagrams of the network of pyramidal cells (P) and basket interneurons (I) which show excitatory (open arrows) and inhibitory (filled arrows) inputs to the pyramidal population. (The widths of arrows indicate relative strength). On the right are plotted excitatory (dashed lines) and inhibitory (solid lines) drives, D, to the pyramidal cells whose concurrent output, P, is shown below. Increases in output, excitation and inhibition are shown as upward deflections. The lines are thickened at times appropriate to the diagrams at the left. Part A shows the state of the pyramidal inputs and outputs just after a stimulus-induced pulse has arrived, causing excitatory input, S, above the background, N. Parts B and C show later points in the development of oscillatory poststimulus pyramidal output.
the pulse. On the right of the figure are plotted excitatory (dotted line) and inhibitory (solid line) drives, D, to the pyramidal cells, and the concurrent pyramidal outputs, P, all as functions of time, T. The plots show the development of an oscillating pyramidal output. Since the model interneurons effectively invert and amplify their input, they push the pyramidal output down to subnormal levels when the signal arrives at time T = 2 (Fig. 8b). That is, pyramidal cells, though still excited by background N are very strongly inhibited by highly driven interneurons, L. Continuing, because of decreased pyramidal output, the interneurons become underdriven at time T = 3 (Fig. 8c), thereby releasing inhibition of the pyramidal cells. The subnormal inhibition, I?, allows the constant excitatory background, iV, to drive the pyramidal output high again (disinhibition). An increase in pyramidal output must be followed by a recurrent increase times until the pulse of activity
in inhibition, and so the cycle may be repeated has dispersed.
several
293
Dispersion in a hippocampal network
5.2 Delayed pyramidal cell excitation The above discussion indicates that network activity levels can oscillate following perturbation anywhere within the network. For example, a transient rise in inhibitory input to the pyramidal population can be followed by a later peak of pyramidal spiking activity. Fornix stimulation in cats causes most pyramidal cells to show long lasting IPSPs and only a few show antecedent EPSPs or antidromic spike potentials [8,9]. Thus, although a portion of the pyramidal population may be immediately excited by fornix stimulation, most cells need not be individually excited by an antidromic action potential over their axons. Wide spatial dispersion of pyramidal-interneuron-pyramidal interactions is assumed to distribute the effects of excitation and tends to lock the pyramidal cells together as a population. Experimental evidence is shown in Fig. 9 consistent with the hypothesis that a pyramidal cell may not be directly excited during the first few milliseconds following fornix excitation, yet may be excited at a later time by disinhibition. Figure 9 shows that a shock stimulus of increasing strength causes progressively longer poststimulus suppressions, D, for an individual cell, eventually followed by a marked peak of excitation, P, which is in turn followed by a second period of depression. This sequence of periods of decreased, increased, and then decreased activity could reflect constraints on such pyramidal cells due to their placement in a neural network. (The AEP deflections (Fig. 9B) slightly lead the PST histogram valleys and peaks, reflecting membrane potential changes prior to spike initiation.) SA A
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Fig. 9. Single-shock pyramidal cell PST histograms (A) as functions of stimulus strength, (B) with nearby AEP. These data are from anesthetized cat hipprocampus (CA3 region) following dorsal fornix stimulation. After a stimulus artifact, SA, a pronounced depression, D, of cell firing probability occurs, followed (after a period depending on stimulus strength) by a rise above the background level to a peak, P, followed by another depression. AEP deflections slightly precede PST deflections. Note the absence of early orthodromic excitation or spike evoked potentials of the cell. There is a significant steady output level of this cell prior to stimulation.
5.3 Network parameter
changes
Increases in stimulus strength or in background excitation are associated with increases in pyramidal oscillation amplitudes; increasing stimulus strength or decreasing
294
J. M. HOROWITZand J. W. B. MATES
background are associated with increases in the interval duration between first and second pyramidal activity peaks. Yet, variations of either of the just-mentioned parameters do not affect the interval durations between later successive peaks (except in the degenerate non-oscillatory cases of zero background). The effects of including dispersion of synchronous activity in simulations of the hippocampal network have not led to significant differences in calculations compared with previous results[6]. For example, the resetting effect of the second shock of a paired-shock stimulus occurs in the present model, as noted in an early model without dispersion and in experimental data [6]. Experimental AEPs and PST histograms have shown progressive flattening of successive peaks of pyramidal oscillations [6, 111. And the simulations show that dispersion of axon diameters and synaptic delays in the hippocampal network can cause a similar progressive flattening of the oscillatory responses to single shocks. In the model a single parameter represents aspects of both delay and dispersion over the neural feedback branch. The effects of changing this parameter, W, are shown in Fig. 6. The increase in the damping and period of simulated pyramidal oscillations with increasing W parallels experimental data from cats. With increasing depth of sodium pentobarbital anesthesia, the oscillations of AEPs became more heavily damped and the interval durations between first and second peaks became longer [ 111. Simulations have shown four ways to increase the duration between the first two peaks in pyramidal cell activity: (1) by increasing stimulus strength; (2) by decreasing excitatory background activity; (3) by increasing the mean feedback delay; (4) by increasing a coupled feedback delay-dispersion. Whereas changes in the first two parameters leave the later intervals unchanged (i.e., constant frequency), changes of the latter two parameters decrement the overall frequency. The experimental measurements suggest that intervals between second and third peaks remain constant, in juxtaposition to the expected effects of (3) and (4). In part because amplitudes of the hippocampal AEPs do not increase with depth of anesthesia, (1) can be rejected. Thus, anesthesia may suppress background excitation of the hippocampal network and leave the loop delay within the network unaffected. 5.4 Pulse train driving The model network increasingly blocks higher frequency pulse train inputs. Regularly occurring bursts of action potentials (with a maximum intraburst frequency reported at 50 Hz) arising from the cells of the medial septum and driving the pyramidal cells of the hippocampus, lead to the 0 rhythm [21,22]. The increasing opacity of the simulated hippocampal network to higher input frequencies suggests that pyramidal neurons may respond as a population only to the envelopes of the driving septal bursts during genesis of the 8 rhythm. On the other hand, were the intraburst frequency low enough, the simulations indicate that both inter- and intraburst timing could be followed, perhaps corresponding to the fast activity observed in hippocampal EEG [26]. Although hippocampal EEGs have a cutoff above about 40 Hz[27,28], enough bandwidth seems available to the hippocampal network to allow its response to individual spikes at these frequencies. Finally, these present simulations (Fig. 7) predict that, following the onset of a pulse-train of frequency too high for the network to pass, the pyramidal output should show an oscillatory rise to a new, higher firing level. Thus, simulations of pulse-train driving the pyramidal cells predict the result of an untried experiment: PST histograms from intracellular records of pyramidal cell responses to high frequency pulse-trains should show transient oscillatory rises to higher firing rates.
Dispersion in a hippocampal network
29.5
The activity of the model network is self-limiting in the several ways described above. Because of dispersion, its response to a transient input is limited in duration. Due to network delays and long-lasting recurrent inhibition, its response to pulse-train driving wanes at higher frequencies. Nevertheless, the hippocampus is a cortex easily provoked into seizure activity [29]. In this connection, Dichter and Spencer [3] showed that a focus of penicillin-induced hyperexcited pyramidal cells was physically encircled by a surround of strongly inhibited pyramidal cells. They suggested that the powerful recurrent pyramidal inhibitory system is more widely spatially dispersed in the pyramidal cell layer than is the recurrent excitatory system. Thus, an excited focus of cells may be expected to show limited spread. On one hand, the model by Dichter and Spencer [4] for interictal spikes in a malfunctioning cortex shows self-limiting spatial spreading of synchronous pyramidal activity; on the other hand, simulation by the present model shows self-limiting of synchronous pyramidal activity in the frequency and time domains. 5.5 Further
development
of the model HM2
The model described here might be extended by simulation of temporal dispersal of input waveforms (as indicated by recent work modeling dispersion of signals in a fiber tract) [19] and by including a positive feedback loop, representing recurrent excitation of pyramidal cells [17]. It cannot, however, be easily merged with the models of Dieter and Spencer [4] and of Kilmer and McLardy [5] for the hippocampus. Both of these models deal with small sets of representative neurons, whereas the model of this report is based on the lumped behaviors of synchronously-driven neural populations. A major step in the development of the model would be the inclusion of spatial parameters in addition to incorporation of input dispersion and recurrent excitation. It should be noted that the neural model presented here has only one nonlinear element (threshold). Nevertheless, its behavior is complex (e.g., the resetting effect of a second shock to the system). Simulations of the model of this report have given testable predictions and are consistent with experiments regarding the shock-induced pyramidal cell activity of the hippocampus. For example, the delayed peak observed in some PST histograms is consistent with the long-term behavior predicted by model HM2. 6. SUMMARY A CSMP model of a hippocampal neural network has been presented. The model represents two lumped populations of hippocampal neurons, the pyramidal cells and interneurons (basket cells). These populations are coupled in a recurrent feedback loop with basket cells inhibiting pyramidal cells and pyramidal cells exciting basket cells. The model is second generation and diff ers from the parent model by inclusion of a dispersion element representing parameter variations within the cells’ populations which tend to disperse otherwise synchronous activity. REFERENCES
1. J. D. Green and A. Arduini, Hippocampal electrical activity in arousal, J. Neurophysiuf. 17,533 (1954). 2. T. L. Bennett, Hippocampat theta activity-a review, Commun. Behav. BioZ. 6, 37 (1971). 3. M. Dichter and W. A. Spencer, Penicillin-induced interictal discharges from cat hippocampus-I. Characteristics and topographical features, J. NeurophysioL 32, 649 (1%9). 4. M. Dichter and W. A. Spencer, Penicillin-induced interictal discharges from cat hippocampus-11. Mechanisms underlying origin and restriction, J. Neurophysiol. 32, 663 (1%9). 5. W. L. Kilmer andT. McLardy, A model of hippocampal CA3 circuitry, Int. J. Neurosci. 1,107(1970).
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6. J. M. Horowitz, W. J. Freeman and P. J. Stoll, The neural network with a background level of excitation in the cat hippocampus, ht. J. Neurosci. 5, 113 (1973). 7. E. R. Kandel, W. A. Spencer and F. J. Brinley, Electrophysiology of hippocampal neurons, J. Neurophysiol. 24, 225 (1961). 8. P. Andersen, J. C. Eccles and Y. Lyning, Location of postsynaptic inhibitory synapses on hippocampal pyramids, J. Neurophysiol. 27, 592 (1964). 9. P. Andersen, J. C. Eccles and Y. Lyning, Pathway of postsynaptic inhibition in the hippocampus, J. Neurophysiol.
27, 608 (1964).
10. W. J. Freeman, Distribution in time and space of prepyriform electrical activity, J. Neurophysiol. 22,644 (1959). 11. J. M. Horowitz, Evoked activity of single units and neural populations in the hippocampus of the cat, Electroenceph.
clin. Neurophysiol.
32, 277 (1972).
12. L. Harmon and E. Lewis, Neural modeling, Physiol. Rev. 46, 513 (1%6). 13. S. Raymon y Cajal, Studies on the Cerebral Cortex (Limbic Structures) (Translated by L. M. Kraft) Lloyd-Luke, London, pp. l-179 (1955). 14. S. Raymon y Cajal, The Structure of Ammon’s Horn (Translated by L. M. Kraft) C. C. Thomas, Springfield, pp. l-78 (1968). 15. R. Lorente De No, Studies on the Structure of cerebral cortex--II. Continuation of the study of the ammonic system, J. Physiol. Neurol. 46, 113 (1934). 16. W. A. Spencer and E. R. Kandel, Hippocampal neuron responses to selective activation of recurrent collaterals of hippocampofugal axons, Expt. Neurol. 4, 149 (1961). 17. R. M. Lebovitz, M. Dichter and W. A. Spencer, Recurrent excitation in the CA3 region of cat hippocampus, Int. .I. Neurosci. 2, 99 (1971). 18. E. R. Kandel and W. A. Spencer, Electrophysiology of hippocampal neurons--II. After potentials and repetitive firing, .I. Neurophysiol. 24, 243 (1961). 19. P. J. Smietana, J. M. Horowitz and T. J. Willey, Dispersion along fiber tracts and in the coupling between tracts and a cortical network. Submitted to Comp. Prog. Biomed. 20. D. P. Artemenko, Role of hippocampal neurons in theta-wave generation, Neurophysiology 4,409 (1973). (Translated from Russian NPHTBl, 4 (3), 343-428, 1972.) 21. H. Petsche, Ch. Stumpf and G. Gogolak, The significance of the rabbit’s septum as a relay station between the midbrain and the hippocampus, Efectroenceph. clin. Neurophysiol. 14, 202 (1%2). 22. G. Apostol and 0. D. Creutzfeldt, Cross-correlation between the activity of septal units and hippocampal EEG during arousal, Brain Res. 67, 65 (1974). 23. P. W. Landfield, J. L. McGaugh and R. J. Tusa, Theta rhythm: a temporal correlate of memory storage processes in the rat, Science 175, 87 (1972). 24. J. M. Horowitz. M. A. Saleh and R. D. Karem. Correlation of hinnocampal theta rhythm with changes in cutaneous temperature. Am. J. Physiol. 227, 635 (1974). __ _ 25. J. V. Wilson and P. R. Burgess, Disinhibition in the cat spinal cord, J. Neurophysiol. 25, 392 (1962). 26. Ch. Stumpf, The fast component in the electrical activity of rabbit’s hippocampus, Electroenceph. clin. Neurophysiol.
18, 477 (1%5).
21. J. C. Boudreau, computer measurements of hippocampal fast activity in cats with chronically implanted electrodes, Electroenceph. clin. Neurophysiol. 20, 165 (1%6). 28. E. Basar and C. Ozesmi, The hippocampal EEG activity and a systems analytical interpretation of averaged evoked potentials of the brain, Kybernetik 12, 45 (1972). 29. J. D. Green, The hippocampus, Handbook of Physiology, Sec. 1. Neurophysiology, Vol. 2, p. 1373, American Physiological Society, Washington, D.C. (l%O). About the Author-JOHN M. HOROWITZ received his M.S. degree in electrical engineering and his Ph.D. degree in biophysics (in 1%8) from the University of California, Berkeley. Dr. Horowitz then joined the staff at the University of California, Davis, and is presently an Associate Professor in the Department of Animal Physiology. His research interests have centered on the neural control of temperature regulation and neural networks in the mammalian nervous system and he has published several articles in these fields. He is a member of the American Physiological Society and the Society for Neuroscience. About the Author-JOHN W. B. MATESreceived the B.A. degrees in Biology and Physics from the University of California at Berkeley in January, 1966. In August 1973, following graduate work in cytogenetics, neurophysiology and ethology, he received a Ph.D from the Biology Department of the University of Oregon with a thesis on chameleon oculomotor behavior. He is particularly interested in using computer techniques for analyses of models and data which bear on the understanding of behavior in evolutionary, ecological and neurophysiological terms. At present, Dr. Mates is a postdoctoral scholar in the Department of Animal Physiology, University of California at Davis.
Pergamon Press1975. Vol.5.pp.X3-2%. Pnnted inGreatBritain
SIGNAL
DISPERSION WITHIN A HIPPOCAMPAL NEURAL NETWORK* J. M. HOROWITZ and J. W. B. MATES
Department of Animal Physiology, University of California, Davis, CA95616, U.S.A. (Received 18 December 1974 and in reaisedforrn 4 March 1975) Abstract-A model network is described, representing two neural populations coupled so that one population is inhibited by activity it excites in the other. Parameters and operations within the model represent EPSPs, IPSPs, neural thresholds, conduction delays, background activity and spatial and temporal dispersion of signals passing from one population to the other. Simulations of single-shock and pulse-train driving of the network are presented for various parameter values. Neuronal events from 100 to 300 msec following stimulation are given special consideration in model calculations. Neural model
Hippocampus
tl rhythm
1. INTRODUCTION Neighboring pyramidal neurons of the hippocampus are oriented in parallel and lie in a single sheet. Macroelectrodes, placed in this sheet of cells, sometimes record a slow (4-7 Hz in the rabbit), high amplitude EEG wave called the “theta rhythm.” The hippocampus and its 8 rhythm are of special interest because they have been related to arousal, learning and memoryll-21. Models of hippocampal neural networks have been put forward in attempts to shed light on hippocampal function and malfunction [3-61. A neural model was previously developedE61 representing a recurrent inhibitory feedback network of hippocampal pyramidal cells and interneurons [7-91. It allowed the simulation of experiments and the comparison of calculated and experimental hippocampal behaviors. Using arguments like those of Freeman[ lo] regarding neural populations with inhibitory feedback and spontaneous activity, damped oscillatory evoked potentials (AE Ps) and poststimulus time (PST) histograms had been expected and were recorded from appropriately stimulated cat hippocampus[ll]. Using a minimum number of parameters [ 121, the hippocampal model’s behavior was consistent with experimentally measured hippocampal responses to single-shock excitation of pyramidal cells. Furthermore, model simulations successfully predicted the hippocampal response to pairedshock stimulation. Factors modifying simulated hippocampal waveforms were studied and were presented together with the results of related experiments[6]. The new hippocampal model described here relaxes an assumption which, although convenient, was not biologically tenable; namely, that signal delays between pyramidal cells and interneurons were all identical. The long-term dispersing effects of relaxing this assumption were unclear. By placement of a convolution in the present model’s feedback branch, these dispersive effects were simulated and examined. Further waveforms were calc:ulated for pulse-train driving and for long-term behavior (i.e. over many hundreds of * This project was supported in part by NASA Grant NGR-05-004-099. by University of California Grant D-529, and by Public Health Service Grant NIMH06686. 383
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milliseconds), inviting comparison of model outputs with hippocampal EEG. Specifically, this report describes the behavior of a model hippocampal network, (1) modified to include dispersive effects, (2) driven by pulse-trains, and (3) observed over long time spans. 2. ANATOMICAL AND PHYSIOLOGICAL BACKGROUND 2.1 Anatomical connections of pyramidal ceils The model neural network represents two populations of hippocampal cells, the pyramidal cells, in a layer below the ventricular surface of the hippocampus, and the basket cells, lying adjacent to the pyramidal cells but closer to the ventricular surface (Fig. 1). Fibers terminating near pyramidal cells have been traced back to other pyramidal cells via Schaffer axon collaterals and the longitudinal association system, to the entorhinal cortex, to the septum via the fornix and to basket cells which synapse directly on the somata of the pyramidal cells in dense basket-like assemblages [ 13-151. Pyramidal cell axons are sent into the fornix, and collaterals of some of these axons synapse with basket cells and form the just-mentioned Schaffer collaterals and longitudinal association system. 2.2 Pyramidal cell electrophysiology
Despite the complexity of connections involving pyramidal cells, relatively simple functional networks have been identified by stimulation of the fornix which both orthodromically and antidromically excites pyramidal cells. Following such stimulation, Spencer and Kandel[l6] recorded long-lasting and powerful IPSPs in individual pyramidal cells. By sectioning the fornix and thus abolishing orthodromic input, they were able to show, by stimulation on the hippocampal side of the section, that the pyramidal cells could be antidromically activated and that the IPSPs were still observed. They inferred the presence of a recurrent inhibitory loop back to pyramidal cells. Andersen et al. [8] demonstrated that the IPSPs were produced at or near the pyramidal cell somata, and suggested that an interneuron was in the recurrent loop and that the interneuron was a basket cell [9]. The somatic location of the synaptic inhibition appears
Fig. 1. Schematic of pyramidal cell-basket cell neural network. Recurrent inhibition of pyramidal cells. P2, via their axon collaterals and basket cells, BA, is indicated by black arrows 1-3, 5-7. Arrow 1 represents input to pyramidal cells, Pl. arriving on fibers from the fornix. Arrow 4, on P l’s axon. represents pyramidal output traveling along the hippocampal surface to the fornix. Arrow 8 represents recurrent excitation of pyramidal cells, P 1, by other pyramidal axon collaterals, P 2.
Dispersion in a hippocampal network
285
placed to maximize the modulatory effect of negative feedback on pyramidal cell output. Additional records obtained from antidromically activated pyramidal cells in such deafferented preparations were recently interpreted as evidence for recurrent excitatory activation of pyramidal cells [17]. Anatomical grounds for such recurrent excitation are available in the Schaffer collaterals and longitudinal association system, and again in the basket cells. Lebovitz[17] has pointed out, however, that the massive recurrent inhibition of pyramidal cells effectively obscures other weaker influences on pyramidal cell output. Whereas most intracellular records of pyramidal cells taken by Andersen et al. [8,9] showed prolonged poststimulus IPSPs, few cells showed short-latency (i.e. less than 5 msec lag) EPSPs or antidromically evoked spike potentials. Signals on axon collaterals of a few pyramidal cells may be sufficient to cause inhibition of many pyramidal cells. Durations of EPSPs and IPSPs were reported to average less than 10 msec and about 300 msec respectively. Even in the absence of applied electrical stimulation of the fornix, EPSPs and IPSPs are assumed to be constantly generated in the network as reflected by the steady-state rates of pyramidal cell spike generation [6,11,18]. And because of the wide dispersion of inhibitory feedback among pyramidal cells, one may expect both pyramidal and basket cell populations to have signficant mean firing activity levels.
ideally
2.3 Hippocampal EEG and behavior Pyramidal cell activity has been linked to the generation of the hippocampal 0 rhythm. For example, hippocampal pyramidal cell spiking activity is usually phaselocked to the 13 waves[20]. However, the drive for the rhythm has been traced to structures outside the hippocampus. Petsche et al. [21] showed that a class of cells in the medial septum fired rhythmic bursts of spikes in step with the hippocampal 0 rhythm; that during post-seizure hippocampal silence, these cells were silent, and that prior to every recurrence of the 13waves, the septal cells were again rhythmically firing bursts of spikes. Cross-correlations measured between septal and hippocampal EEGs and between septal unit activity and hippocampal EEGs show that the activity of certain septal cells (the “B” cells) closely correlates with the amplitude of the 8 rhythm, and that hippocampal EEGs lagged those of the septum[22]. Occurrence of the 8 rhythm in hippocampal EEGs has been correlated with alertingtl], with memory storage[23] and with a variety of other behaviorally-related events. The 8 rhythm can be induced by novel stimuli such as would tend to arouse an animal (e.g. light flashes, handclaps); cutaneous temperature transients were also found to evoke 0 activity in the unanesthetized, loosely restrained rabbit [24], adding yet another sensory modality to those previously shown to elicit this rhythm. 3. HIPPOCAMPAL MODEL HM2 The model HM2 is an extension of one described earlier 161,and is diagrammed in Fig. 2. (The properties of the pyramidal and interneuron populations selected for modeling are overlaid on outlines of a pyramidal and basket cell.) The signals passing through an element in the network represent averages over all the cells and fibers corresponding to that element. Provided one assumes the average poststimulus activity of any single neuron is the same as that of all other neurons of its type, one can relate experimental
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PYRAMIDAL
CELL
Fig. 2. The neural population model in block form. Pyramidal cell components include two EPSP elements, an IPSP element, and a nonlinear element T. A DISPERSE and DELAY element lie on the feedback branch which includes the basket cells. N represents steady background EPSP activity from INPUT,,. After a delay, a second element simulates EPSPs elicited by shock (impulse) stimulations delivered to input fibers at INPUT,. If the summed EPSPs and IPSP exceed a threshold, the non-linear T element delivers an output. The delayed and dispersed output over the feedback branch inhibits pyramidal cells.
data in the form of PST histograms of single pyramidal cells to calculated model outputs[6]. A special advantage of this formulation is that it allows the use of continuous signals in the model simulating the network. In Fig. 2, three inputs are shown summed by the modeled pyramidal cell population: (1) EPSPs following excitation of a fiber tract (at INPUT,); (2) EPSPs from a steady background excitation N (from INPUT,); and (3) IPSPs recurrently fed back onto the pyramidal population by interneurons presumed to be basket cells. Shock stimulation of fibers at INPUT, is assumed to cause a rectangular pulse (PULSE element of Fig. 2) of excitatory input to the pyramidal cell population after a delay (DELAY element at left of Fig. 2). This input causes depolarizations or EPSPs of the pyramidal cell membranes. If the sum of the EPSPs and IPSPs exceeds a fixed threshold, the pyramidal population produces a non-zero output level (T element, Fig. 2); otherwise the output is zero. The output excites interneurons (basket cells) which, after a delay, (right-most DELAY element, Fig. 2), inhibits the pyramidal cells by causing membrane hyperpolarizations or IPSPs. The DISPERSE element of Fig. 2 represents the lumped effect of individual cell parameter variations causing dispersion of otherwise synchronous activity around the feedback loop. This element convolves its input with a dispersing function to produce its output. Figure 3 is a flow diagram of the computation sequence corresponding to the blocks of Fig. 2. Background pyramidal excitation, N, is held constant during each simulation run. The stimulus-induced EPSPs and IPSPs are characterized: EPSP = x,
(1)
IPSP = -n * y, d.3 + k,i + c,x =
excitatory
(2)
input signal,
(3)
287
Dispersion in a hippocampal network
Fig. 3. Formulation of model HM2 for CSMP programming. The set of boxes with open circles in their upper left corners are for calculating poststimulus EPSP transients; boxes with filled circles are for calculating IPSP transients. *d&per (convolution) requires a Fortran subroutine as does the nonlinear T (threshold) element.
d,j + k,j + c,y = inhibitory input signal.
(4)
The constants c,, c,, d,, d,, k,, k,, are chosen so that the EPSPs and IPSPs induced by pulse inputs are overdamped transients which resemble experimentally recorded intracellular potentials [8,9] (In addition, the equations can be related to state space variables for transmitter release)[6]. The number of pyramidal cells excited by a single basket cell is proportional to n, a parameter in the feedback branch and in equation (2) for IPSPs. The non-linear element, T, behaves according to: OUTPUT=D-T where
for
DzT;
OUTPUT = 0.0
for
Dee T;
(5)
D = EPSP + IPSP + N.
T represents a critical population firing threshold analogous to that of an individual ceil. Temporal dispersion of the signals in the feedback loop is calculated as the following convolution by the DISPERSE element (Fig. 2) whose input and output at time, t, are FL(t) and F,+,(t), respectively where disper(t) is set to zero for t 50. FL+,(t) =
I
+= am
-r
*disper (t - t’) - dt’
compactly, F L+l = FL*disper.
(6)
The integral is approximated stepwise by means of serial products[l9]. For the simulations of this report, the “disper” function was rectangular with width, W. The total loop delay equals W/2 units plus the pure loop delay (DELAY element), so variations in
C.E
M. 5/bC
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288
W will cause variations in the total loop delay. Without compensating normalization of the “disper” function’s area, changes of W cause changes in the loop gain. The DELAY elements of Figs. 2 and 3 represent pure time delays. Multiplicative elements of Fig. 3 amplify signal levels by a fornix gain, gi, and a loop gain, gl. The PULSE element of Fig. 3 produces a rectangular output pulse in response to an input impluse. Equations (l-6), together with the flow diagram in Fig. 3, describe model HM2. The major assumption in the model is a particular connection of cell populations based on experimental data outlined in sections 2.1 and 2.2. 4. RESULTS The model is shown in Fig. 3 as formulated for the Continuous Systems Modeling Program (CSMP) described in IBM’s users manual H20-0367-1. Constants and inputs used in the simulations (shown in Fig. 3) described in this report are presented in Table 1. Figures 4-7 show simulated pyramidal output traces which are assumed comparable with experimentally acquired PST histograms accumulated from poststimulus spiking records of individual pyramidal cells. For all simulated output traces shown in this report (refer to the labeled trace A of Fig. 4), the first input signal (the leading edge of a pulse) arrived at the EPSP element immediately prior to the rise of the first peak, P,, of pyramidal output. A depression of pyramidal activity, D, (extending below the mean firing level by a displacement, &), was usually followed by a peak of pyramidal activity, Pz (extending above the previous resting level of pyramidal output by P,“). Each trace Table 1. Constants and parameters used in CSMP calculations of waveforms INPuTa:
Pulse height = 1.0; pulses Of Figure Input gain, Of Figure Farnix
for trace
A
= 0.520.
Background activity, N = 0.004.
INPuTb :
EPSP and
(except gi=2000).
gi=looo 4 where
delay
(except =O.OlO).
“idal=0.015 7 where width
IPSP:
X8
=
xb
=
dx = d
Cx kx
FEEDBACK LOOP:
Y = -700; =
y,
=
yb
=
0.0.
1.0. c
-
-200.
-loooooY k = -10000 Y
Threshold Loop gain, Dispersion
of s1
T element =
10;
= 0.0. lcmp delay
= 0.025.
fmmction, ‘disper,’ is rectangular viei width = 25,
heigh - 0.04. Fan O"t, " = moo.
Dispersion in a hippocampal network
Fig. 4. Calculated pyramidal waveforms as parameter N, representing background activity, was decreased (top to bottom). For N = 0 (trace E) no oscillations occur. With decreasing N the second peak, Pz, moves slightly away from the first, P,; but the later interval durations do not vary. Dots are placed 5 msec apart.
Fig. 5. Calculated pyramidal waveforms as parameter PF, representing delay in the feedback loop, was decreased (top to bottom). The arrows indicate the second poststimulus pyramidal activity peaks. Oscillation periods decrease with decreasing delay. Very short delays cause rapid damping of the waveform (bottom two traces).
289
290
J. M.
HOROWITZ and J.
-.r,j
s
A
W. B.
5
S
p3
MATES
w
=I00
w=40
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pz
5
w= IO
CA d
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4 \
p2
p3
w=2
Fig. 6. Calculated pyramidal waveforms as parameter W, representing the width of the normalized rectangular dispersing function in the feedback loop, was decreased (top to bottom). The waveform period decreases with decreasing W (because of decreasing total loop delay).
shown in the present figures represents 500 msec of continuous activity; the points on each trace are 5 msec apart and computer calculations were made in steps corresponding to 1 msec intervals. Finally, all traces shown start after initial transient activity of the network had declined to a steady level. Simulated pyramidal outputs following a single stimulus pulse are traced in Figs. 4-6 as various model parameters are altered. Figure 4 shows the effects of variations of the background activity, N, upon the pyramidal output. Variation of N has no effect on the long term oscillation frequency; although, with decreasing N, the time between first and second output peaks (P, and P,) increases slightly. Increasing N causes increased oscillation amplitude. With no background activity (N = 0), no oscillations occur. Figure 5 shows the effects of variation in the feedback loop delay. The longer the delay, the longer the period of the oscillatory output. Very short delays cause damping of pyramidal output. Feedback loop dispersion, W, is varied in Fig. 6. The heights of the rectangular “disper” functions were altered to compensate for changes of their widths (i.e. the rectangular dispersing function with which the feedback signal is convolved was normalized). Increasing W increases the total loop delay which causes the oscillation period to increase. Figure 7 shows simulated pyramidal output initiated by pulse-train driving at four different frequencies. At the lowest driving frequency (trace A), the network can pass each pulse to the output: but at the highest driving frequency (trace D), the network responded only to the initial step-like transient of mean excitation (by an oscillatory rise
Dispersion in a hippocampal network
291
Fig. 7. Calculated pulse-train driven pyramidal waveforms as functions of increasing driving frequency (top to bottom). For each trace, the first pulse of the train arrived at the EPSP element just prior to the first pyramidal activity peak. The impulse frequency at INPUT, (Figs2,3) was varied. Top to bottom, the impulse frequencies were IOHz, 50 Hz, 200 Hz and 1000 Hz, respectively. In response to each impulse, however, the PULSE element (Figs 2,3) generated a pulse of width 10 msec. Thus pulses arrived at the EPSP element at 10 Hz, 50 Hz, 66 Hz and 99 Hz. Pyramidal cell output level increases as the mean input increases (10 msec pulses at 66 Hz and at 99 Hz, lower two traces); note the poor high frequency response.
of pyramidal output to a higher steady level), and was unable to pass the input frequency to its output. Thus, the neural model behaves as a low pass filter, blocking high input frequencies and passing low input frequencies. 5. DISCUSSION 5.1 Oscillatory activity PST histograms for hippocampal pyramidal cells commonly display a succession of peaks[ll]. Prior to the shock excitation, pyramidal cells are modeled as firing at a constant rate. Therefore the simulated network is in an active state (except when background activity N is set to zero [Fig. 41). There are two available means of increasing simulated pyramidal output: (1) activity of input excitatory fibers can drive pyramidal firing frequency higher; (2) by contrast, suppression of inhibitory feedback allows normal excitatory activity to drive the firing frequency upwards. This latter case has been oblserved in the spinal cord and called “disinhibition” by Wilson and Burgess [25]. Peaks in model pyramidal cell activity, corresponding to peaks observed in oscillatory PST histograms, can thus be due either to added excitatory inputs or to disinhibition. Figure 8 illustrates how oscillatory network activity may develop following single pulse excitation of pyramidal cells (P). Parts a, b and c of the figure represent three states of the model network at successive times 1,2 and 3 in a continuum of states initiated by
J. M. HOROWITZand J. W. B. MATES
292
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-42 ‘,F
c
plILz!IL 0
1
T
2
I
-4 9
p*
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I
2
3
T
13
Fig. 8. Schematic “snapshots” of pyramidal inputs and outputs at three successive points in time. Parts a, band c represent times T = 1, T = 2 and T = 3, in the poststimulus history of the network. On the left are diagrams of the network of pyramidal cells (P) and basket interneurons (I) which show excitatory (open arrows) and inhibitory (filled arrows) inputs to the pyramidal population. (The widths of arrows indicate relative strength). On the right are plotted excitatory (dashed lines) and inhibitory (solid lines) drives, D, to the pyramidal cells whose concurrent output, P, is shown below. Increases in output, excitation and inhibition are shown as upward deflections. The lines are thickened at times appropriate to the diagrams at the left. Part A shows the state of the pyramidal inputs and outputs just after a stimulus-induced pulse has arrived, causing excitatory input, S, above the background, N. Parts B and C show later points in the development of oscillatory poststimulus pyramidal output.
the pulse. On the right of the figure are plotted excitatory (dotted line) and inhibitory (solid line) drives, D, to the pyramidal cells, and the concurrent pyramidal outputs, P, all as functions of time, T. The plots show the development of an oscillating pyramidal output. Since the model interneurons effectively invert and amplify their input, they push the pyramidal output down to subnormal levels when the signal arrives at time T = 2 (Fig. 8b). That is, pyramidal cells, though still excited by background N are very strongly inhibited by highly driven interneurons, L. Continuing, because of decreased pyramidal output, the interneurons become underdriven at time T = 3 (Fig. 8c), thereby releasing inhibition of the pyramidal cells. The subnormal inhibition, I?, allows the constant excitatory background, iV, to drive the pyramidal output high again (disinhibition). An increase in pyramidal output must be followed by a recurrent increase times until the pulse of activity
in inhibition, and so the cycle may be repeated has dispersed.
several
293
Dispersion in a hippocampal network
5.2 Delayed pyramidal cell excitation The above discussion indicates that network activity levels can oscillate following perturbation anywhere within the network. For example, a transient rise in inhibitory input to the pyramidal population can be followed by a later peak of pyramidal spiking activity. Fornix stimulation in cats causes most pyramidal cells to show long lasting IPSPs and only a few show antecedent EPSPs or antidromic spike potentials [8,9]. Thus, although a portion of the pyramidal population may be immediately excited by fornix stimulation, most cells need not be individually excited by an antidromic action potential over their axons. Wide spatial dispersion of pyramidal-interneuron-pyramidal interactions is assumed to distribute the effects of excitation and tends to lock the pyramidal cells together as a population. Experimental evidence is shown in Fig. 9 consistent with the hypothesis that a pyramidal cell may not be directly excited during the first few milliseconds following fornix excitation, yet may be excited at a later time by disinhibition. Figure 9 shows that a shock stimulus of increasing strength causes progressively longer poststimulus suppressions, D, for an individual cell, eventually followed by a marked peak of excitation, P, which is in turn followed by a second period of depression. This sequence of periods of decreased, increased, and then decreased activity could reflect constraints on such pyramidal cells due to their placement in a neural network. (The AEP deflections (Fig. 9B) slightly lead the PST histogram valleys and peaks, reflecting membrane potential changes prior to spike initiation.) SA A
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Fig. 9. Single-shock pyramidal cell PST histograms (A) as functions of stimulus strength, (B) with nearby AEP. These data are from anesthetized cat hipprocampus (CA3 region) following dorsal fornix stimulation. After a stimulus artifact, SA, a pronounced depression, D, of cell firing probability occurs, followed (after a period depending on stimulus strength) by a rise above the background level to a peak, P, followed by another depression. AEP deflections slightly precede PST deflections. Note the absence of early orthodromic excitation or spike evoked potentials of the cell. There is a significant steady output level of this cell prior to stimulation.
5.3 Network parameter
changes
Increases in stimulus strength or in background excitation are associated with increases in pyramidal oscillation amplitudes; increasing stimulus strength or decreasing
294
J. M. HOROWITZand J. W. B. MATES
background are associated with increases in the interval duration between first and second pyramidal activity peaks. Yet, variations of either of the just-mentioned parameters do not affect the interval durations between later successive peaks (except in the degenerate non-oscillatory cases of zero background). The effects of including dispersion of synchronous activity in simulations of the hippocampal network have not led to significant differences in calculations compared with previous results[6]. For example, the resetting effect of the second shock of a paired-shock stimulus occurs in the present model, as noted in an early model without dispersion and in experimental data [6]. Experimental AEPs and PST histograms have shown progressive flattening of successive peaks of pyramidal oscillations [6, 111. And the simulations show that dispersion of axon diameters and synaptic delays in the hippocampal network can cause a similar progressive flattening of the oscillatory responses to single shocks. In the model a single parameter represents aspects of both delay and dispersion over the neural feedback branch. The effects of changing this parameter, W, are shown in Fig. 6. The increase in the damping and period of simulated pyramidal oscillations with increasing W parallels experimental data from cats. With increasing depth of sodium pentobarbital anesthesia, the oscillations of AEPs became more heavily damped and the interval durations between first and second peaks became longer [ 111. Simulations have shown four ways to increase the duration between the first two peaks in pyramidal cell activity: (1) by increasing stimulus strength; (2) by decreasing excitatory background activity; (3) by increasing the mean feedback delay; (4) by increasing a coupled feedback delay-dispersion. Whereas changes in the first two parameters leave the later intervals unchanged (i.e., constant frequency), changes of the latter two parameters decrement the overall frequency. The experimental measurements suggest that intervals between second and third peaks remain constant, in juxtaposition to the expected effects of (3) and (4). In part because amplitudes of the hippocampal AEPs do not increase with depth of anesthesia, (1) can be rejected. Thus, anesthesia may suppress background excitation of the hippocampal network and leave the loop delay within the network unaffected. 5.4 Pulse train driving The model network increasingly blocks higher frequency pulse train inputs. Regularly occurring bursts of action potentials (with a maximum intraburst frequency reported at 50 Hz) arising from the cells of the medial septum and driving the pyramidal cells of the hippocampus, lead to the 0 rhythm [21,22]. The increasing opacity of the simulated hippocampal network to higher input frequencies suggests that pyramidal neurons may respond as a population only to the envelopes of the driving septal bursts during genesis of the 8 rhythm. On the other hand, were the intraburst frequency low enough, the simulations indicate that both inter- and intraburst timing could be followed, perhaps corresponding to the fast activity observed in hippocampal EEG [26]. Although hippocampal EEGs have a cutoff above about 40 Hz[27,28], enough bandwidth seems available to the hippocampal network to allow its response to individual spikes at these frequencies. Finally, these present simulations (Fig. 7) predict that, following the onset of a pulse-train of frequency too high for the network to pass, the pyramidal output should show an oscillatory rise to a new, higher firing level. Thus, simulations of pulse-train driving the pyramidal cells predict the result of an untried experiment: PST histograms from intracellular records of pyramidal cell responses to high frequency pulse-trains should show transient oscillatory rises to higher firing rates.
Dispersion in a hippocampal network
29.5
The activity of the model network is self-limiting in the several ways described above. Because of dispersion, its response to a transient input is limited in duration. Due to network delays and long-lasting recurrent inhibition, its response to pulse-train driving wanes at higher frequencies. Nevertheless, the hippocampus is a cortex easily provoked into seizure activity [29]. In this connection, Dichter and Spencer [3] showed that a focus of penicillin-induced hyperexcited pyramidal cells was physically encircled by a surround of strongly inhibited pyramidal cells. They suggested that the powerful recurrent pyramidal inhibitory system is more widely spatially dispersed in the pyramidal cell layer than is the recurrent excitatory system. Thus, an excited focus of cells may be expected to show limited spread. On one hand, the model by Dichter and Spencer [4] for interictal spikes in a malfunctioning cortex shows self-limiting spatial spreading of synchronous pyramidal activity; on the other hand, simulation by the present model shows self-limiting of synchronous pyramidal activity in the frequency and time domains. 5.5 Further
development
of the model HM2
The model described here might be extended by simulation of temporal dispersal of input waveforms (as indicated by recent work modeling dispersion of signals in a fiber tract) [19] and by including a positive feedback loop, representing recurrent excitation of pyramidal cells [17]. It cannot, however, be easily merged with the models of Dieter and Spencer [4] and of Kilmer and McLardy [5] for the hippocampus. Both of these models deal with small sets of representative neurons, whereas the model of this report is based on the lumped behaviors of synchronously-driven neural populations. A major step in the development of the model would be the inclusion of spatial parameters in addition to incorporation of input dispersion and recurrent excitation. It should be noted that the neural model presented here has only one nonlinear element (threshold). Nevertheless, its behavior is complex (e.g., the resetting effect of a second shock to the system). Simulations of the model of this report have given testable predictions and are consistent with experiments regarding the shock-induced pyramidal cell activity of the hippocampus. For example, the delayed peak observed in some PST histograms is consistent with the long-term behavior predicted by model HM2. 6. SUMMARY A CSMP model of a hippocampal neural network has been presented. The model represents two lumped populations of hippocampal neurons, the pyramidal cells and interneurons (basket cells). These populations are coupled in a recurrent feedback loop with basket cells inhibiting pyramidal cells and pyramidal cells exciting basket cells. The model is second generation and diff ers from the parent model by inclusion of a dispersion element representing parameter variations within the cells’ populations which tend to disperse otherwise synchronous activity. REFERENCES
1. J. D. Green and A. Arduini, Hippocampal electrical activity in arousal, J. Neurophysiuf. 17,533 (1954). 2. T. L. Bennett, Hippocampat theta activity-a review, Commun. Behav. BioZ. 6, 37 (1971). 3. M. Dichter and W. A. Spencer, Penicillin-induced interictal discharges from cat hippocampus-I. Characteristics and topographical features, J. NeurophysioL 32, 649 (1%9). 4. M. Dichter and W. A. Spencer, Penicillin-induced interictal discharges from cat hippocampus-11. Mechanisms underlying origin and restriction, J. Neurophysiol. 32, 663 (1%9). 5. W. L. Kilmer andT. McLardy, A model of hippocampal CA3 circuitry, Int. J. Neurosci. 1,107(1970).
2%
J. M. HOROWITZand J. W. B. MATES
6. J. M. Horowitz, W. J. Freeman and P. J. Stoll, The neural network with a background level of excitation in the cat hippocampus, ht. J. Neurosci. 5, 113 (1973). 7. E. R. Kandel, W. A. Spencer and F. J. Brinley, Electrophysiology of hippocampal neurons, J. Neurophysiol. 24, 225 (1961). 8. P. Andersen, J. C. Eccles and Y. Lyning, Location of postsynaptic inhibitory synapses on hippocampal pyramids, J. Neurophysiol. 27, 592 (1964). 9. P. Andersen, J. C. Eccles and Y. Lyning, Pathway of postsynaptic inhibition in the hippocampus, J. Neurophysiol.
27, 608 (1964).
10. W. J. Freeman, Distribution in time and space of prepyriform electrical activity, J. Neurophysiol. 22,644 (1959). 11. J. M. Horowitz, Evoked activity of single units and neural populations in the hippocampus of the cat, Electroenceph.
clin. Neurophysiol.
32, 277 (1972).
12. L. Harmon and E. Lewis, Neural modeling, Physiol. Rev. 46, 513 (1%6). 13. S. Raymon y Cajal, Studies on the Cerebral Cortex (Limbic Structures) (Translated by L. M. Kraft) Lloyd-Luke, London, pp. l-179 (1955). 14. S. Raymon y Cajal, The Structure of Ammon’s Horn (Translated by L. M. Kraft) C. C. Thomas, Springfield, pp. l-78 (1968). 15. R. Lorente De No, Studies on the Structure of cerebral cortex--II. Continuation of the study of the ammonic system, J. Physiol. Neurol. 46, 113 (1934). 16. W. A. Spencer and E. R. Kandel, Hippocampal neuron responses to selective activation of recurrent collaterals of hippocampofugal axons, Expt. Neurol. 4, 149 (1961). 17. R. M. Lebovitz, M. Dichter and W. A. Spencer, Recurrent excitation in the CA3 region of cat hippocampus, Int. .I. Neurosci. 2, 99 (1971). 18. E. R. Kandel and W. A. Spencer, Electrophysiology of hippocampal neurons--II. After potentials and repetitive firing, .I. Neurophysiol. 24, 243 (1961). 19. P. J. Smietana, J. M. Horowitz and T. J. Willey, Dispersion along fiber tracts and in the coupling between tracts and a cortical network. Submitted to Comp. Prog. Biomed. 20. D. P. Artemenko, Role of hippocampal neurons in theta-wave generation, Neurophysiology 4,409 (1973). (Translated from Russian NPHTBl, 4 (3), 343-428, 1972.) 21. H. Petsche, Ch. Stumpf and G. Gogolak, The significance of the rabbit’s septum as a relay station between the midbrain and the hippocampus, Efectroenceph. clin. Neurophysiol. 14, 202 (1%2). 22. G. Apostol and 0. D. Creutzfeldt, Cross-correlation between the activity of septal units and hippocampal EEG during arousal, Brain Res. 67, 65 (1974). 23. P. W. Landfield, J. L. McGaugh and R. J. Tusa, Theta rhythm: a temporal correlate of memory storage processes in the rat, Science 175, 87 (1972). 24. J. M. Horowitz. M. A. Saleh and R. D. Karem. Correlation of hinnocampal theta rhythm with changes in cutaneous temperature. Am. J. Physiol. 227, 635 (1974). __ _ 25. J. V. Wilson and P. R. Burgess, Disinhibition in the cat spinal cord, J. Neurophysiol. 25, 392 (1962). 26. Ch. Stumpf, The fast component in the electrical activity of rabbit’s hippocampus, Electroenceph. clin. Neurophysiol.
18, 477 (1%5).
21. J. C. Boudreau, computer measurements of hippocampal fast activity in cats with chronically implanted electrodes, Electroenceph. clin. Neurophysiol. 20, 165 (1%6). 28. E. Basar and C. Ozesmi, The hippocampal EEG activity and a systems analytical interpretation of averaged evoked potentials of the brain, Kybernetik 12, 45 (1972). 29. J. D. Green, The hippocampus, Handbook of Physiology, Sec. 1. Neurophysiology, Vol. 2, p. 1373, American Physiological Society, Washington, D.C. (l%O). About the Author-JOHN M. HOROWITZ received his M.S. degree in electrical engineering and his Ph.D. degree in biophysics (in 1%8) from the University of California, Berkeley. Dr. Horowitz then joined the staff at the University of California, Davis, and is presently an Associate Professor in the Department of Animal Physiology. His research interests have centered on the neural control of temperature regulation and neural networks in the mammalian nervous system and he has published several articles in these fields. He is a member of the American Physiological Society and the Society for Neuroscience. About the Author-JOHN W. B. MATESreceived the B.A. degrees in Biology and Physics from the University of California at Berkeley in January, 1966. In August 1973, following graduate work in cytogenetics, neurophysiology and ethology, he received a Ph.D from the Biology Department of the University of Oregon with a thesis on chameleon oculomotor behavior. He is particularly interested in using computer techniques for analyses of models and data which bear on the understanding of behavior in evolutionary, ecological and neurophysiological terms. At present, Dr. Mates is a postdoctoral scholar in the Department of Animal Physiology, University of California at Davis.
