synchronized afterdischarges in the hippocampus: simulation studies of the cellular mechanism

A’euroscience Vol. 12, No. 4, pp. 119i-1200, 1984 Printed in Great Britain

~30~4522/84 $3.00+ 0.00 Pergamon Press Ltd

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SYNCHRONIZED AFTERDISCHARGES IN THE HIPPOCAMPUS: SIMULATrON STUDIES OF THE

CELLULAR MECHANISM

R. D. TRAUB*~, W. D. KNOWLES*?, R. MILE@ and R. K. S. WONG~ *IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A.; tNeurologica1 Institute, New York, NY 10032, U.S.A.; IDepartment of Physiology and Biophysics, University of Texas Medical

Branch, Galveston, TX 77550, U.S.A.

Abstract-Synchronized multiple bursts represent an epileptic neuronal behavior transitional between synchronized single bursts (interictal spikes) and self-sustained seizures. As described in the previous paper, synchronized multiple bursts occur in hippocampal slices treated with picrotoxin. Multiple bursts consist of an initial prolonged depolarizing burst followed by a rhythmi~l series of afterdischarges. Both the initial burst and the afterdischarges are synaptically elicited.

Our previously described model of the interictal spike illustrates that the generation of a single synchronized burst requires a neuronal network possessing the following properties: intrinsic bursting capability of individual neurons, the presence of recurrent excitatory connections between principal neurons and the blockade of synaptic inhibition. The model demonstrates that the generation of single synchronized bursts involves the initial excitation of one or more neurons, and the subsequent sequential spread of excitation through a population of neurons via recurrent excitatory synapses.

In the present study. we examined whether this same mechanism assumed in the previous model could also allow for the generation of synchronized afterdischarges in a population of neurons. We tested the effects of manipulating three network factors: synaptic strength, synaptic density and the refractoriness in the population members following a period of excitation. We discovered that the refractory period following prolonged excitation assumed in our previous model was insufficient to allow for afterdischarge generation. Once sufKcient refractoriness was introduced, afterdischarges appeared in our network of neurons. In the present study, the required refractoriness was attributed to the properties of pyramidal cell axons. In principle, such refractoriness might be located elsewhere in the network. The possible contribution of axonal properties is emphasized because of the known intermittent conduction in other axons. Our present model also reproduced other ex~~mental data. Thus, if the network was too small or if synaptic strength was too small, then only a single synchronized burst occurred. The basic assumptions of this model are both biologically plausible and experimentally testable.

When the hippocampal slice is bathed in convulsant agents, synchronized neuronal bursting occur~~‘~~*~~ that is analogous to the cellular activity underlying interictal spikes in the electroencephalogram of some patients with epilepsy. In the hippocampal slice, synchronized bursts originate in the CA2-CA3 region3*+ which contains pyramidal cells that burst spontaneously (asynchronously) even in the absence of convulsant agents.42.4’ Such agents (e.g. penicillin or bicuculline) block synaptic inhibition mediated by y-aminobutyric acid (GABA).‘2,‘“,4’ We have previously shown how the known excitatory synaptic interconnections22.2~,26 can lead to a long-latency single population burst (as observed experimentally) if we assume these connections to be sparse and random.38,39 We used random connections in these previous studies because of the absence of anatomical data on the local excitatory synaptic recurrents.

The previous paper27 has described how, in the presence of the GABA-bl~ker picrotoxin (0.1 mM), a more complex form of neuronal population activity occurs in all pyramidal cell areas of the hippocampal _-...____

Abbreriution: GABA, ;-aminobutyric acid.

slice. Whereas the application of penicillin results in single synchronized bursts, multiple synchronized bursts are observed after exposure to picrotoxin. The experimental observations from CA3 pyramidal cells most relevant to this simulation study are as follows: multiple bursts consist of a prolonged initial burst lasting up to 150 ms followed by a series of afterdischarges. The interval between successive afterdischarges is 45-65 ms (15-22Hz). All pyramidal cells appear to participate in multiple bursts, which are highly synchronized. Intracellular depolarizations of individual neurons are coincident with the locally recorded field potential, and are synchronous with each other in paired intracellular recordings. A hyperpolarizing afterpotential usually appears following the final afterdischarge. However, the intracellular hyperpolarization which typically follows a single burst is frequently missing or replaced by a depolarizing afterpotential which underlies the series of afterdischarges. This depoiari~ng afterpotential, however, is not necessarily by itself the cause of the afterdischarges. Chemical synaptic interactions are required for the generation of afterdischarges as well as the initial synchronized burst.

1191

1192 R. D. Traub et al.

The synaptic reversal potentials associated respectively with the primary burst and with the afterdischarges are the same to within experimental error.27 On switching to a high Mg2+-low Ca2+ solution, afterdischarges are blocked one by one, leaving the initial synchronized burst intact, before it too is finally blocked. In addition, synaptic inputs into a given cell occur rhythmically and in phase with discharges from the rest of the population. As pre- dicted by our previous mode1,)8*39 synchronized activity may be influenced by a single neuron.” In local regions of the CA2-CA3 area, rhythmic popufation activity is partially entrained and its timing reset by the intracellular stimulation of a single neuron in approximately one-third of neurons tested.‘”

Understanding in vitro afterdischarges is relevant to studies of in uivo experimental tonic seizures and also to human patients with tonic fits and myoclonus. The cellular events in the in vitro afterdischarges appear to resemble the events observed by Mat- sumoto and Ajmone Marsan” in penicillin-induced in vivo neocortical ictal phenomena and by Sawa et al.29 with in viva electrically induced seizures. Both groups observed rhythmical field potentials, every 50-100 ms, that were coincident in time with cellular depolarizations in at least some of the neurons. Such phenomena may underlie the IO-25 Hz epileptic re- cruiting rhythm or paroxysmal fast activity recorded in the electroencephalogram of some patients with tonic seizures, and in the runs of electro- encephalographic spikes recorded in some patients with generalized tonic-clonic epilepsy.b~‘o~‘4~‘s

Our previous work3* has demonstrated that three factors are critical for the occurrence of the single synchronized epileptiform burst elicited by penicilhn and other GABA-blocking agents: (1) intrinsic bursting, (2) recurrent synaptic excitation and (3) synaptic disinhibition. The synchronization mechanism described in our previous paper” is supported by two important experimental observations: stimulation of a single cell is capable of eliciting a synchronized population burst, and a burst in one cell is able to evoke a burst in a synaptically connected follower ce11.26 One prediction of that model is this: there is a cascading sequential recruitment of neuronal bursting into the population response. If refractoriness develops in different neurons in a corresponding sequential way, then it seems plausible that residual activity from one synchronized burst could initiate a second synchronized burst, and so on in series.

The goal of the present study was therefore to determine whether our previous network model of penicillin-induced epileptiform synchronizatior?” would also generate synchronized multiple bursts. To do this, we examined the effects of changing three factors that define the interactions between cells: synaptic connection density, synaptic strength and refractoriness in interceIlu~ar communication. It was investigated whether neurons intrinsically contain suficient refYa.ctoriness for a series of afterdischarges

to be generated by the mechanism described above. It was further considered whether the occurrence of some form of absolute refractoriness following a period of excitation is critical for the appearance of epileptiform afterdischarges, and whether a pattern of intermittent conduction of axonal firing accounts for some of this refractoriness. Such refractoriness has been described elsewhere. for example in the feline spinal cord.’ The presence of recurrent axonal branching may be important for the occurrence of intermittent conduction. Additional studies of in- vertebrate axons suggest that axonal branch points represent a region of low safety factor for action potential propagation.‘6.“.“” Under conditions of rapid sustained repetitive firing, a variety of factors. including extracellular K ’ accumulation may con- tribute to conduction failure. By including the effect of axonal refractoriness, we attempted to reproduce all of the crucial experimental data, using reasonable network parameters: 100 cells, an average of 2.5 inputs/cell, and an average synaptic delay of 5 ms.

EXPERIMENTAL PROCEDURES

In our multi-compartment model of a CA3 hippocampal neuron,” bursting consists of a series of fast Na -mediated spikes, riding on a depolarizing wave. followed by a slow, largely Ca’” -mediated spike. The burst IS terminated by two events: (1) a K + current (Rowing through conductance “g,“) that is activated by intracellular accumulation of Paz+ ions and that decays with slow exponential time course and (2) by inactivation of ,Q~. There are abdundant data to suggest that a CaLi-mediated K- current exists in hip~ocampa~ neurons. This current is activated by repetitive firing,” epi~eptiform bursting in CA I neurons.7 spontaneous bursting in CA3 neurons” and ~i~r~?toxin-induced bursting in CA3 neurons.” The single ceil model reproduces the phenomenology of bursting. but the ionic currents are not described with quantitatrve accuracy.”

We began with preliminary smulations of single model neurons and of synaptically coupled pairs of model neurons. Synaptic inhibition was not included. because of the biock- ade of such inhibition by picrotoxin.:,?” We did not postulate any effect of picrotoxin on intrinsic membrane properties, although we are aware that picrotoxin may increase the intrinsic excitability of spinal cord neurons.’

In the first set of simulations. synaptic coupling in neuronal networks was achieved as before.” When the membrane potential of any cell soma was depolarized 20 mV or more relative to rest. an output signal was fed into a delay line, representing conduction and synaptic delays, that led to the other cells to which the original cell was connected. When this signal reached the “postsynaptic” cell. a 60 mV battery through a pre-determined fixed synaptic conductance (0.05-I .O ~4s or O.9- I X.0 mS!cm’) was opened across the membrane. Delays of 3.5. SOms were tested. although in most simulations delays of about 4-S ms were used. Synaptic inputs were to the same dendritic compartments as in the previous study lx (0.25 i, from the soma in apical dendrites, 0.15 and 0.25 i. from the soma for basilar dendrites). The total synaptic conductance was distributed between the four different compartments in proportion to their respective membrane areas. Synaptic conductances opened by two simultaneous inputs from ditrerent “presynaptic” cells were added linearly. In order to avoid artificial synchronizing effects caused by having cells with identical intrinsic properties interconnected through identical delay lines, we randomly scattered the synaptic delays in some simulations. Thus. whether one ceil was lo receive

Cellular mechanism of afterdischarges 1193

input from another cell was determined by one random choice (employing a pseudo-random number generator), and the duration of the delay (if there was a connection between the two cells) was determined by a second random choice, using a uniform distribution over some pre-specified interval.

For comparison with the above parameter values, we quote some recently published data’ on mossy fiber inputs to CA3 cells in the presence of GABA-blocking agents. In the study of Brown and Johnston,9 a small (but unknown) number of granule cells were stimulated and recordings were made under voltage clamp from CA3 somas. They estimated an excitatory postsynaptic potential reversal of 60-70mV positive to resting potential, an excitatory postsynaptic potential conductance of about 0.02 nS, and a time to peak synaptic conductance about 6 ms after stimulation.

In order to include the effect of axonal refractoriness, communication between cells worked in the following way. A signal s(t) is defined to be 1 if both the soma voltage l’(t) 2 20 mV relative to rest, and also dY/dr 2 50 V/s. ,9(t) is defined to be 0 otherwise. Input into the axon thus follows soma action potentials rather than plateau depolarizations. Prel~inarv unoublished data of R. Miles and R. K. S. Wong from the hippocampal slice suggest that this may be a more accurate representation of axonal output: individual action potentials in a presynaptic neuron lead to individual inhibitory postsynaptic potentials in a monosynaptically connected postsynaptic neuron. However, presynaptic action potentials occurring at rates faster than 150 Hz are not followed 1:l by postsynaptic potentials. In our model, s(t) is usually equal to 1 for 0.25-0.3 ms during each simulated soma action potential. We define an axon refractoriness variable R(r) for each cell as follows: R(O)= 0, and dR/df = s(t) - R(t)/z& Thus, R(t) increases as it receives axonal signals [when .S(t) = I] and decreases exponentially with time constant rR. A single action potential increases R by about 0.25. We now allow transmission along the axon if s(t) = 1 (a signal is entering the axon) and R(f) 5 RThrerh, where RThmh is a threshold value for the refractoriness

Al 7-.-L..--

125 mV

Fig 1. Simulations of axonal output for different values of the refractoriness parameters rR and RThErh. A 100ms 0.25 FS conductance pulse (Al) elicits in an isolated neuron a burst followed by a steady train of action potentials (A2), about 6 ms apart. The response of the axon to this somatic signal is shown in (B) for different values of the refractoriness parameters. (Bt) rcR = 30 ms, Rtie.,, = 1.5 (usual values). (B2) T, = 30 ms, R,,,, = 1.75. (B3) T* = 30 ms. R Thrrsh = 1.25. ‘(B4) rll =%-ms, R,,,, = l:j. 035) tR = 40 ms, Rnrrsh = 1.5. In each case, the axon passes two closely spaced pulses during the initial burst, followed by a pause. The number of action potentials transmitted during the repetitive train depends on the parameter choices for the axon. In (Bl), the last two somatic action potentials are not

conducted.

variable. The two parameters zR and RThmh define this refractoriness behavior. Typical values are ?R = 30 ms and R Thrrsh = 1.5. Axonal outputs corresponding to a particular somatic burst for some different choices of the parameters rR and Rrhrrrh are illustrated in Fig. 1. If a signal *‘passes” through the axon, it enters a delay line as described above. Upon arrival at the postsynaptic membrane, it also acts as described above (connecting a 60mV battery through a synaptic conductance), but with one difference: instead of acting for only one time step, the signal acts for ten time steps (0.5ms). This smoothing is necessary for numerical stability of the integration program, since synaptic inputs are now “choppier” than in the original model.

Preliminary simulations with two or more cells interconnected by excitatory synapses often led to a state in which all cells became depolarized, with the synaptic connections maintaining this state once it had developed. Such a state is not surprising in a system with positive feedback. This state was not a problem in our previous study3s since, unlike the present case, we were not there concerned with generating a self-sustaining series of oscillations, and we were thus able to use relatively small synaptic strengths. In order to inhibit the onset of this excited state, we chose, in most simulations, to enhance the early phase of the burst afterhyperpolarization by adding to the single cell model a slow voltage-dependent K+ current (an M-current). Such a current is known to exist in hippocampal neurons1.20 and has been studied by voltage clamp techniques in these cells and in bullfrog sympathetic ganglion cells.7~20 It is also known that the burst afterhyperpolarization in hippocampal neurons has at least two components’s Voltage clamp studies indicate that (I) the peak M-current is small relative to the fast voltage-dependent Hodgkin-Huxley KC current; (2) the steady state of g, as a function of voltage is sigmoidal and (3) rM is smaller near VK (the K equilibrium potential) than at more depolarized potentials. In accord with this, we let & = 0.1 gK, and let g, = j&,x, where x is the state variable for this current. We let I, = g,( V - V,) and we chose txY( V) = 0.02/{ 1 + exp[(40 - V)/S]} and j,(V) = 0.01 exp[( 17 - k’)/ 181 (these are the forward and backward rate functions, respectively). With these rate functions, rX at the K equilibrium potential (- 15 mV with respect to resting potential) was 16.9 ms, at 19 mV positive to resting potential it was 108 ms, and at 29 mV positive to rest it was 140 ms. Halliwell and Adams3 measured the time constant for closure of M-channels at 19 mV positive to resting potential as 47ms and at 29 mV positive to rest as 92 ms. These measurements were at 30°C. Another indication that the time constants used in the model are probably of the right order of magnitude for a hippocampal neuron is the following: the burst afterhyperpolarization observed by Hotson and Prince” in CA1 neurons in the presence of Mn2+ ion lasted about 5Oms. Mn’+ should block both Ca*+-de~ndent K currents and synaptic activity, and thus uncover any voltage-dependent intrinsic hyperpolarizing currents. (An early component of the afterhyperpolarization following a burst is also resistant to Mg2+ ion in CA1 cells.‘*) The rate functions we used give a sigmoidal shape to the curve of x, vs voltage (not shown). Inclusion of this M-current in the model does not significantly alter the appearance of a single current-induced burst (not shown). Likewise, the basic synaptically elicited synchronization process described previously38 remains intact after inclusion of the M-current. There is a quantitative difference pro- duced by including the M-current: in order to produce double bursts in certain cells, one must use larger synaptic strengths so as to overcome the additional post-burst afterhyperpolarization.

In order to allow for the possibility that the M-current kinetics described above are too slow, we also performed a number of simulations in which the M-current kinetics were speeded up two-fold (i.e. a,(V) and fl,( vf were multiplied by two). In addition. in some simulations, the M-current was

1194 R. D. Traub et al.

effectively removed by decreasing its maximum conductance 40-fold.

Differential equations were integrated with a second order Taylor series method” with time step 50 p.s. PL-I programs were run on an IBM 3081 digital computer. Field potentials were estimated as in the previous study.“*

RESULTS

Our basic problem was to determine if a randomly connected network of nerve cells could generate the following type of behavior: stimulation of one or a few cells should lead, via synaptic interactions, to a long synchronized burst succeeded by a rhythmical series of shorter synchronized bursts, in all of the

cells. Although excitatory synaptic actions have been

demonstrated between pyramidal cells in

CA2-CA3,24.2h no direct evidence is available con-

cerning the pattern of these connections. We considered interconnection patterns that were random, since, as noted above, the actual pattern is not known, and since we wished to avoid imposing a particular artificial structure on the system. Some general features of random networks are discussed in the Appendix. Two properties of random networks relevant to epilepsy emerge from that analysis: (I) a number of loops of different lengths are likely to exist, allowing re-entrant paths for the continuation of a series of afterdischarges, and (2) the number of cells which can be reached starting from a given cell may include most of the population. This latter feature determines whether a local stimulus can “grow”, via synaptic interactions. into a population

event. We discovered that in a given random network,

increasing synaptic strength could lead to an in-

creased duration of the synchronized burst (Fig. 2). so that this burst came to resemble the primary bursts observed experimentally. Note that with larger synaptic strengths, the input to a given cell is not only larger at each time but also lasts longer. This can only happen because of cooperative interactions between the cells. If synaptic strength was large enough, cells would exhibit a long depolarizing tail with rapid runs of spikes and synaptic potentials superimposed (Fig. 2). These tails resemble similar events seen rarely in penicillin-induced interictal events’” and in cultured hippocampal neurons.” With larger synaptic strengths (as for Fig. 2D), two types of afterdischarges were seen (not shown). In small networks (e.g. 20 cells), a high frequency train of action potentials appeared following the initial synchronized burst. In sufficiently large networks (100 cells). a rhythmical synchronized series of afterdischarges occurred, but each afterdischarge lasted 1 O&200 ms, with a period between afterdischarges of about 200 ms. This period is much longer than what is observed experimentally.

We observed (Fig. 3) that increasing network connectivity tended to make the initial synchronized

A

B

c

D

I 50 mV

100 ms Fig. 2. Effect of increasing synaptic strength on duration of synchronized burst in a random network. Four simulations (A-D) were done, each using the same network of 100 cells, with each cell having 2.5 inputs on the average. The stimulus was in each case a 2.5 nA injected current, beginning at time 0, for 100ms into four cells. Plotted in each case are the soma membrane potential of one fixed cell (that has inputs from two other cells), and the synaptic input into that cell (below). Synaptic strengths were 0.2 (A), 0.4 (B), 0.5 (C) and 0.6 (D) hSS/input. Synaptic delay was 5ms. Note that as synaptic strength is increased, the synchronized burst (corresponding to the primary burst) becomes Ionger. The vertical

calibration represents 3.3 PS synaptic input.


SYNAPTIC STRENGTH

strong Weak

----_-I 100 Ins

Fig. 3. Effect of increasing network connectivity on synchronized burst with “strong” synapses (0.2pS-A) and “weak” synapses (0.05 pS-B). The networks have 100 cells and the stimulus is 2.5 nA for 100 ms injected into four cells. Plotted in each case is the soma membrane potential of one cell. Synaptic delay is 5ms in each case. Connectivities (expressed as expected number of inputs/cell) are: 2.5 (line 1) and 5.0 (line 2). With the “strong” synapses (but not the “weak” ones), increasing the connectivity prolongs cellular

bursting.

100

A

B

Cl

burst longer, without by itself leading to after-

discharges. This lengthening effect was marked with sufficiently large synaptic strengths, but was minimal with synapses that were weaker [albeit still strong enough to lead to a synchronized population event (Fig. 3)]. We found, as in our original study3* (Fig. 2A3 of that paper), that a few cells would develop a double burst, if the synaptic strength was adequate.

Such cells tended to be ones whose initial burst occurred early relative to the rest of the population. The tendency toward double bursting was not noticeably enhanced by increasing the size of the network from 49 to 400 cells (not shown).

When the effects of axonal refractoriness were included (see Experimental Procedures), using rea-

sonable network parameters (100 cells, 2.5

inputs/cell, synaptic delay 3.5-5.5 ms), we were able to simulate a rhythmical series of synchronized bursts with an appropriate frequency (Fig. 4). Specific axonal refractoriness variables are rR = 30 ms and

R Thresh = 1.5. Note the sustained initial burst and the

Dl

D2

50 mV

100 ms Fig. 4. Synchronized afterdischarges in a model that includes axonal refractoriness. There are 100 cells, an average of 2.5 inputs/cell, synaptic delays are randomly scattered between 3.5 and 5.5 ms, synaptic strength is 0.3~s. The refractoriness parameters are rR = 30ms, RThmsh = 1.5. The stimulus was an intracellularly injected current of 2.5 nA for 100 ms into four cells (not shown), starting at time 0. The figure illustrates the number of cells with soma membrane potential 2 20 mV relative to resting potential (A), estimated field potential (B-positive up), intracellular soma membrane potentials of two cells (Cl and Dl) and their respective synaptic inputs (C2 and D2). The cell of(C) receives input from four other cells, while the cell of(D) receives input from two other cells. Note the rhythmical oscillations with period about 75 ms. Synchronization is relative rather than absolute. The voltage calibration refers to intra-

cellular potentials.

R. D. Traub et rii.

50 mV

100 ms

Fig. 5. Decreasing synaptic strength progressively abolishes simulated afterdischarges. The simulation of Fig. 4 was repeated with all parameters identical except that synaptic strength was gradually reduced from 0.235 /IS (A) to 0.22 /IS (B), 0.20 FS (C), and 0.1 pts (D). Shown are the soma membrane potential of the same cell as in Fig. 4(C) (upper traces) and the synaptic input to that cell (lower traces). As synaptic

strength is reduced, the initial burst is also shortened, as happens experimentally.

periodic afterdischarges. The period in this case is 75ms, at the high end of what is observed experimentally. The graph of the number of cells with soma potential 220 mV (Fig. 4A) indicates that synchronization is pronounced but not absolute. Thus, there are always a few cells firing that can serve to keep the process going, in spite of the fact that synaptic delays are only about 5 ms. Synaptic input into each ceil is in phase with the afterdischarges themselves, as observed experimentafly.27 The oscil- latory behavior illustrated in Fig. 4 does not depend, in the model, on the existence of the M-current, since it persists when g, is reduced 40-fold (not shown).

In order to confirm that the two refractoriness variables were not chosen in a completely arbitrary way, we explored the effects of varying them (not shown). The results of varying these parameters were in accord with physical intuition. If tR is increased to 40 ms, the qualitative behavior of the system is little changed, but if it is increased to 50 ms, only a synchronized double burst occurs without afterdischarges. The physical significance of this is that if axonal refractoriness takes too Iong to recover, the system will “die” before the first afterdischarge can be initiated. On the other hand, if zR is decreased to 20ms, the afterdischarge consists of almost con- tinuous action potentials with brief interruptions. Thus, having refractoriness recover too quickly re- sembles having no refractoriness at all. Next, Rnrrsh was varied while 7R was kept fixed at 30 ms. If RThiesh was increased to 1.75 (from 1.5), an almost con- tinuous afterdischarge occurred. This makes sense:

with RThresh too large, the axon is less refractory and the system behavior comes to resemble the behavior of the system without any refractoriness. On the other hand, if RThresh is reduced too far (e.g. to 0.79, then only a single synchronized burst occurs: the axon blocks after only a few action potentials. We conclude that to reproduce the experimentally observed afterdischarges, the axon should pass about 6-12 closely spaced action potentials before begin-

1

1 I lhlll

SO rn”

Fig. 6(A). A network that is too small does not support afterdischarges. The simulation of Fig. 5(C) (which has one afterdischarge) was repeated, with all parameters the same except for size of the network. The network here contains only 25 cells. Each cell produces onty a double burst. A representative cell is shown (membrane potential above, synaptic input below). Experiment? indicate that within a sufficiently small piece of the CA2-CA3 region, it may be impossible to elicit afterdischarges in the presence of picrotoxin, even though s~chronized bursting occurs. (B) Simultaneous stimulation of all the cells does not lead to afterdischarges. The simulation of Fig. 5(A) was again repeated, with all network parameters identical. The only change was that now the 2.5 nA 100 ms stimulus previously injected into only four cells was instead injected into all the cells. Two representative cells (B1 and B2) are illustrated (the same cells as in Fig. 4C and 4D). Again, there are no

afterdischarges, but rather double bursts in each cell.


ning to become refractory (see Ex~rimental Pro- cedures), and recovery from refractoriness should proceed with time constant 30-4Oms.

The effects of reducing synaptic strength, while keeping all other parameters constant, are shown in Fig. 5. Note the gradual disappearance of the afterdischarges and the shortening of the initial synchronized burst, in agreement with experimental data.27 The simulation of Fig. S(D) also confirms that with smaller synaptic strengths, and using axonal refractoriness (as described in Experimental Pro- cedures), we are still able to reproduce single synchronized bursts, as occur in penicillin-perfused hippocampal slices, This is an important technical detail, since our original mode138 did not include axonal refractoriness.

duration of a burst (Figs 2 and 3). Increasing synaptic strength and connation density in general tends to prolong the synchronized burst. If synaptic strength is large enough in the model, and there is no axonal refractoriness, then either a rapid continuing train of action potentials follows a single synchronized burst, or else a slow series of synchronized afterdischarges occurs.

When the size of the network is reduced to 25 cells (Fig. 6A), instead of the expected single afterdischarge, there are no afterdischarges. This behavior was confirmed using a larger synaptic strength as well (0.3 pS--not shown). The explanation is as follows. In a large enough system, there are always at least a few cells that are depolarized (as in Fig. 4A). With a sufficiently small system, if there is any synchronization of the initial burst, this feature will disappear. The result is that there is not enough residual activity to initiate the first afterdischarge. Experiments presented in the companion paper2’ also indicate that su~ciently small pieces of the CA2-CA3 region do not support afterdischarges, even when they may be able to support single synchronized bursts. Consistent with this notion, if all the model cells are stimulated simultaneously (causing them to repolarize almost simultaneously), then afterdischarges again do not occur (Fig. 6B). This is the case even in a network with 100 cells. Thus, as in Fig. 6(A), the process “dies” after the initial synchronized burst. It has not yet proven possible to test this concept experimentally.

DISCUSSION

It was our hope that we could find a biologically reasonable set of parameters with which a random network of neurons could generate a series of afterdischarges resembling those described in the companion paper. 27 In order to accomplish this, it was necessary, in the second part of our study, to add to the model an additional physiological effect: refractoriness in the transmission of a signal from presynaptic soma to postsynaptic dendritic membrane. The clue that such a mechanism might work came from observing a sustained series of action potentials when synaptic strength was sufficiently large in some simulations. This suggested that if communication between cells was interrupted at times, oscillations of the system might occur. Although we have called this refractoriness process “axonal”, the model does not distinguish where along the path from presynaptic to postsynaptic cell the refractoriness is actually located (e.g. axon initial segment, branch points, presynaptic terminals, etc.). Our hypothesis of axonal refractoriness seems plausible in view of the behavior of other known systems,4*‘6s’7,36 although of course it needs direct experimental confirmation in the hippocampal slice. We note that conduction failures may develop at sites of axonal recurrent branching, and also that recurrent axonal connections are apparently the essential excitatory pathways by which hippocampal neurons are recruited into synchronized epileptiform events. We have described refractoriness in the simplest phenomenological way we could devise, using only two parameters (zR and RThresh). A reasonable range of these two parameters supports the desired model behavior, i.e. the occurrence of synchronized afterdischarges at a frequency close to what is observed experimentally.

This study consists of two basic parts. In the first By incorporating this hypothesis of axonal refrac- part, we studied in detail the effects of manipulating toriness and using experimentally plausible network the network parameters in a model of randomly parameters, we were able to reproduce three crucial interconnected neurons (all connections excitatory). experimental observations:27 (1) the occurrence, after The network parameters are the synaptic strength a localized stimulus, of a long synchronized burst and the connection density. The stimulus in each case followed by a rhythmical series of synchronized was to one or four cells. We confirmed the robustness secondary bursts; (2) the occurrence of only a single of our original model of the synchroni~tion process synchronized burst, with the same stimulus, after underlying the single epileptiform burst, in the sense synaptic strength is reduced and (3) the requirement that the behavior of the system is insensitive to small for a su~ciently large neuronal ensemble in order to changes in the parameters. The specific results of produce a series of afterdischarges (as opposed to these simulations can be summarized as follows. For producing just a single synchronized burst). We a randomly connected network of neurons to gener- reiterate that this remarkable behavior is generated in ate a “primary burst”, i.e. a sustained synchronized the model solefy by excitatory chemical synaptic burst, then the network must have sufficiently strong interactions between cells randomly connected by synaptic connections of sufficiently high density, and “faulty” transmission lines. Each burst in the series have a synaptic delay that is short relative to the is terminated by two factors: intrinsic hyperpolarizing

1198 R. D. Traub ?t al

conductances (both Ca’+-dependent and voltage- dependent in the present model), and interruption of the excitatory input from other ceils. Our model predicts four phenomena: (1) the existence of slight phase differences in the afterdischarges of different cells; (2) the absence of afterdischarges when a strong stimulus simultaneously excites all of the cells. thus leaving no residual activity; (3) the requirement for a critically large number of cells to keep the process “alive” during periods when most of the cells are

initiate a succeeding synchronized burst) and (4) the dropping off, one by one, of the afterdischarges when synaptic strength is reduced, this phenomenon arising because of accumulated neuronal refractoriness after periods of excitation.

This study demonstrates that testable hypotheses

are now available concerning the most fundamental question in the pathophysiology of epilepsy: how do neurons interact so as to produce a seizure?

either below threshold or have axons which are not conducting (so that residual activity from one syn-

Ackno~ledgeme~f.s--Supported in part by NIH grant

chronized burst can spread throughout the popu- NS18464 and a Klingenstein fellowship award (to R.K.S.W.). We wish to thank Drs R. Linsker, F. A. Dodge.

lation via excitatory synaptic ~onnectjons so as to Jr., P. E. Seiden and L. Schulman for helpful discussions.

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19.

20.

21.

22.

23.

24.

25.

26.

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(Accepted 27 January 1984)

APPENDIX

Because random networks are used so extensively in this paper, it may be useful to consider certain aspects of their structure. Networks, or “graphs” as they are sometimes called, are an important class of math~ati~l objects for the understanding of neuronal systems, if we can (I) identify cells with nodes in the network and axons and synapses with directed edges in the network and (2) ignore, to some extent at least, the intrinsic properties of the cells. Some intuitive feel for the complexity of even a sparsely connected random network can be obtained from Fig. 7, which contains 100 nodes, each having an average of 2.5 inputs. We have suggested that the series of afterdischarges arises in part because of the existence of loops. It is therefore of particular importance to examine whether random networks have loops of a given length, and if so, how many such loops. One may estimate the expected number of loops of a given short length (say 2 or 3) by using probabilistic formulae. For example if there are a total of N cells and the probability that cell i connects to cell j is p, then the probability that i and j connect to each other (i.e. form a 2-100~) is pZ. The probability that i forms part of a 2-100~ with some cell other than itself is then 1 - (1 -p’)“. These methods lead to formulae of bewildering complexity with longer loops in networks of finite size, because of the problem of avoiding counting a given eel] twice during the calculation. An alternative approach is to construct a sample random network of the desired size (N = 1000 is a reasonable size for problems related to the hippocampal slice) and then to compute explicitly its loop structure. This is easily done (in principle) by constructing the adjacency matrix-A of the network.95 A is an N-by-N matrix of O’s and 1’s such that A(& j) = 1 if i sends an output toj, and A(i,j) = 0 otherwise. Suppose we look at the diagnonal elements of Ak, the kth matrix power of A. These elements represent the number of

different loops of length k of the given element to itself. Thus, by constructing successive powers of A (using sparse matrix techniques), we can examine all the loop structure of the matrix. The results of ~rfo~ing these calculations for some sparsely connected networks of 1000 cells are shown in Table 1.

Several features are of note in this table. First, some short length loops are present even in sparsely connected networks. Second, there is wide dispersion in the lengths of different loops. This has the consequence that a random network is far from having the kind of cyclic structure that will ensure a uniform delay for re-entrant paths. Finally, a random network with only 5 inputs/cell on the average (which is quite sparse if there are 1000 nodes) has an extremely rich loop structure, in the sense that over 90?< of cells lie on loops of several different lengths.

Another relevant feature of random networks that is of interest is the following: starting at a particular node, how many nodes can be “reached” from it by a path of arbitrary length? This is important in interpreting the data on eliciting a population response by intracellular stimulation of a single cell. It has been shown elsewhere** that about one-third (10 out of 36) cells in CA3 are capable, upon intra~llular simulation, of entraining a series of multiple bursts. We expect that for a cell to have this property, there must be a synaptic path from it to most or all of the other cells in the system. We examined this property in random networks of 500 cells. Such networks are surprisingly strongly connected. Thus, if each cell has 5 inputs on average, then 497 of the cells are connected to 491 or more other cells. With 2.5 inputs/cell, then 447 of the cells connect to 431 or more other cells. Even with 1.8 inputs/cell on average, 376 cells connect to 361 or more cells. The connectivity of the network, measured in this

1200 R. D. Traub el ul.

CELLS -

I I

Fig. 7. A random graph with LOO nodes (“cells”), shown layered across the middle of the diagram, each node having an average of 2.5 inputs. Because it was not possible to draw arrows on the connections, the connections are laid out so that they always run clockwise (in the direction of the large arrows).

Table I. Loon structure of some random ermhs with 1000 nodes

No. cells lying on at least I loop of length k

No. ceils on at least 1 loop of lennth

k=2-

k=3

k=4 k=5 k=6 k=l k=a k=9 k = 10

Expected no. 2.0 2.5

6 6 12 9 37 37 25 54 90 145

180 281 269 473 384 598 495 682 581 682

mputs~cell 3.0 4.0 5.0

6 6 20 IX 68 123 78 217 437

182 485 819 379 791 964 605 924 979 712 943 979 827 945 979 850 945 972 853 94.5 979

way, declines with decreasing the number of inputs/cell. This happens largely because of the occurrence of increasing numbers of cells with either no inputs or no outputs.

We suggest that the CA3 region of the slice may not be “wired” in a random pattern for the following reasons: on the one hand, the CA3 region is densely enough connected for synchronized population events to occur after a localized stimulus, while at the same time only about one-third of ceffs examined experimentally can influence the rest of the population,26 i.e. only about one-third of cells have paths to most of the rest of the network. On

the other hand, a random network with only 1.8 inputs/cell on the average (i.e. “loosely connected”) has the property that over half of the nodes have paths to most of the other nodes. Perhaps the outputs from selected cells have wider distribution or greater efficacy than the outputs from other cells. It is also possible that the in oiuo CA3 region is wired randomly (at least locally), but that non-randomness is introduced by the slicing procedure. Cutting the slice might affect the connections of cells near the faces of the slice in a manner different than connections of cells in the interior of the slice. Such a concept should be testable experimentally.

synchronized afterdischarges in the hippocampus: simulation studies of the cellular mechanism

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