Bulletin o1 Mathematical Bio/o~]y Vo[. 53, No. 4. pp 505 523, ~991. Printed in Great Britain. I

THE THALAMOCORTICAL EPILEPSY

0092-8240/91 $3.00 + 0.00 Pergamon Press pie 1991 Society l'or Malhematical Biology

CONTRIBUTION

TO

WILLIAM J. NOWACK

Department of Neurology University of Arkansas for Medical Sciences Little Rock, AR 72205, U.S.A. GEORGE C. THEODORIDIS

Division of Biomedical Engineering University of Virginia Charlottesville, VA 22908, U.S.A. and University of Patras Patras, Greece The experimental literature has dealt intensively with the cortical contribution to epilepsy. Possibly because of the direction of technological advance, much less attention has been paid to the role of other structures. A model whxch emphasizes the role of some of those non-cortical structures, specifically that of thalamocortical modulation of cortical excitability, is developed. Some aspects of the petit real seizure, a seizure type considered by some investigators to involve thalamocortical mechanisms, are predicted by the model. Although the thalamocortical mechanisms under study are not the only mechanisms underlying seizures, a full understanding of the phenomenology of epilepsy needs to take into account the role of subcortical modification of cortical activities in addition to other mechanisms. Gloor has described two types of epileptogenesis: type I characteristic of non-convulsive seizures and type II characteristic of convulsions. There is disagreement as to whether or not the two mechanisms represent qualitatively different phenomena. Utilizing the thalamocortical model, it can be shown that the two types of epdeptogenesis are qualitatively different. Furthermore, the thalamocortical model leads to a possible explanation of clinically different profiles of antiepileptic efficacy of medications.

1. Introduction. The conceptual framework underlying much epilepsy research has developed out of one of two models. The "epileptic neuron" model seeks the abnormality underlying epilepsy in the properties of the individual neuron. The "epileptic aggregate" theory postulates that the neurons involved in the epileptic discharge may be completely normal while the neurons are so connected that the resulting activity is abnormal. Current research suggests that, rather than being mutually exclusive concepts, both types of abnormality contribute to the development of an epileptic seizure (Fromm, 1987b). In recent years the brain slice preparation has, for technical reasons, been the basis of much of the experimental work into the neurophysiology of epilepsy (Jeffreys and Roberts, 1987). The elegant experimental and simulation work of 505

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Miles et al. (1988) is limited to neurons in a portion of simulated central nervous system equivalent in depth to the hippocampal slice. The brain slice model has led in recent years to a considerably greater understanding of the physiology of epilepsy but has tended to restrict consideration to events occurring in neuronal structures a few hundred microns thick (Jeffreys and Roberts, 1987). In earlier epileptological investigations the role of subcortical mechanisms was actively investigated. Some authors feel that the very successes of the brain slice model have led to overconcentration upon cortical mechanisms (broadened in the last few years to include both neocortex and hippocampal archicortex) and relative minimization of the possible role of subcortical pathways in convulsant or anticonvulsant actions (Gale and Browning, 1988). Engel has speculated that the entire brain is abnormal in epileptics and that investigations into epilepsy should focus on widespread neuroanatomical circuits (1987). Although epilepsy is primarily a disorder of telencephalic cortical structures (Prince and Connors, 1986), the possible contributions of interactions of neurons belonging to different central nervous system structures with different functions to epileptogenesis need also to be examined. Gloor (1979) has defined two types of epileptogenesis, Type I and Type lI. Type I, characterized by the spike-and-wave complex typical of nonconvulsive epilepsy, is based on observations of feline generalized penicillin epilepsy and is most applicable to petit mal or non-convulsive seizures. Type II, characterized electrophysiologically by the focal epileptic spike or sharp wave and clinically by the generalized tonic-clonic seizure, is based largely on observations of the acute penicillin focus and is most applicable to focal epilepsy and by extension to most convulsive epilepsies. The early years of modern electrophysiology were characterized by two theories regarding the mechanism of the generalized spike-wave discharge (GSW): the diffuse cortical hypothesis of Gibbs and Gibbs and the centrencephalic hypothesis of Penfield and Jasper (Avoli, 1988). The diffuse cortical hypothesis localized the primary pathology in cortical dysfunction, although in initial formulations a secondary role for the thalamus was postulated (Gibbs and Gibbs, 1952). The centrencephalic theory attributed generalized "idiopathic" epileptogenesis to a "final level of integration" which included the mesencephalon and diencephalon (Penfield and Jasper, 1954). Later reformulations of the theories by enthusiastic protagonists postulated more of a dichotomy than was actually present (Marcus and Watson, 1966; Gastaut, 1969; Ajmone-Marsan, 1969). In intracranial recordings from children with petit real epilepsy, Williams (1953) found that a three-per-second rhythm begins in the thalamus and spreads to involve the cortex and concluded that petit mal epilepsy represents a thalamic disturbance which then induces widespread rhythmic discharge throughout the cortex. In later work he

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modified his position, feeling that the evidence suggested that epilepsy did not arise in the thalamus but that thalamic structures were important in the propagation of epileptic activity (Williams, 1965). Experimental models were influenced by the theoretical position taken and experimental evidence interpreted as supporting each theory was found (Gloor, 1984; Avoli, 1988). Gloor attempted to reconcile the two theories in his corticoreticular theory, in which GSW is a manifestation of aberrant interaction between cortical and subcortical gray matter (1968). Further identification of the abnormal interaction was forthcoming after the development of a model of GSW, feline generalized penicillin epilepsy (Gloor, 1984). In earlier formulations of the theory, cortex rendered pathologically hyperexcitable responds to thalamocortical volleys which ordinarily produce spindles by generating GSW (Gloor, 1979). Subsequent studies in decorticate cats supported the idea that the thalamic trigger might be necessary but that the development of GSW depended primarily upon changes in the excitability of the cortex (Avoli and Gloor, 1982). Work by Gloor's group utilizing direct extracellular recording from thalamic and cortical neurons provided evidence for a more active role for thalamic neurons in GSW. Thalamic and cortical neurons become tightly locked in rhythmic oscillations between excitation and inhibition. Neither thalamus nor cortex alone appear sufficient to produce or maintain GSW (Avoli et al., 1983). During the tightly coupled rhythmic oscillation in the thalamo-cortico-thalamic loop Gloor feels that the question of whether thalamus drives cortex or cortex drives thalamus is not particularly relevant to the questions under consideration (Gloor, 1984). The question may not even be answerable in terms of a single structure responsible for epilepsy. It is not the intent of this paper, by focusing on the thalamic contribution to epilepsy, to deny the importance of cortical and possibly other subcortical contributions or to rekindle a dispute which is no longer fruitful and has been largely laid to rest, but to indicate other investigative avenues which might yeld additional insights into the question of epilepsy. Changes in thalamocortical excitability are not usually sufficient to give rise to human epilepsy, although stimulation of subcortical structures in experimental animals easily elicits seizures (Fromm, 1987b). Kusske (1976) found that thalamus and cortex independently participate in focal seizures and postulated a role for the thalamus in the transition from purely focal epileptiform activity to generalized epileptiform activity. Noebels and Prince (1978) found that thalamocortiCal axon terminal bursting participated in the development and propagation of focal seizures in cats. Chatt and Ebersole (1988) have shown that the thalamocortical circuitry of cortical layer 4 is essential for the propagation of local epileptiform activity. Elimination of the thalamus can prevent a focal epileptiform discharge from becoming generalized (Aquino-Cias and Bureg, 1967). Miller et al. (1989) report a neuronal

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system in the medial thalamus which regulates both seizures and arousal. While these observations do not prove the active involvement of the thalamus in epileptic activity, in a model of generalized epilepsy it appears reasonable to try to evaluate the role of the thalamus. Emphasis upon the thalamus does not exclude the possibility of other subcortical structures participating in epileptiform activity. With a model which complements those of homogeneous populations of cortical neurons, we evaluate the role which subcortical structures, specifically the thalamus, play in the genesis of generalized epileptic seizures. Previous theoretical models of epilepsy analyzed the behavior of a population of randomly connected neurons in an unstructured net, the "neuronal netlet' or "neuron gas" model (Anninos et al,, 1970; Wong and Harth, ]973). Such models have yielded conclusions regarding the cortical changes associated with epilepsy (Anninos and Cyrulnik 1977; Kokkinidis and Anninos, 1985) and other cortical central nervous system phenomena such as the alpha rhythm (Anninos and Zenone, 1980). Anninos et al. (1970) have derived a formalism, the macroscopic coupling matrix, to deal with the analysis of neural systems which require explicit consideration of more than one netlet but largely postponed consideration of such cases to future investigations. The dynamics of a system consisting of more than one netlet may be complex and mathematically relatively complicated. Two steps in the direction of the complexity of real neural systems have been the examination of the behavior of a netlet under steady external afferent input (Harth et al., 1970) and the subdivision of a single netlet into two distinguishable, interacting subpopulations (Kokkinidis and Anninos, 1985). We further analyze ways in which interactions between cortical and subcortical neurons can lead to epileptic activity. 2. Model. Beurle (1956) viewed the proportion of neurons firing at a given time in a given location as a product of the proportion of cells firing times the residual excitation in cells receiving the stimuli. The model of Wilson and Cowan (1972) incorporates a similar assumption. Assume that the probability p(x, t) of a cortical neuron firing in response to a stimulus of strength x passing through the thalamus at time t is given by a similar product:

p(x, t ) = C(x, t)Th(~, t). C(x, t) can be thought of as the conditional probability of the cortical neuron firing if it receives a thalamocortical stimulus. Angel (1967) has shown that the cortical response to the second of a pair of peripheral stimuli is different from the cortical response to the first stimulus in the pair because of alterations in thalamic transmission. One of the neurophysiological functions of the ventrolateral thalamic nucleus which Purpura (1970) described is that of

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gating, preventing some stimuli from the periphery from reaching the cortex. The Scheibels (1973) have described a neuroanatomic network by which the thalamus could modulate stimuli coming in from the periphery. Th(x, t) can be thought of as the probability that a stimulus is transmitted through the thalamus. In their study of barbiturate spindles, Andersen et al. (1967a,b) found that, while the amplitude of certain cortical thalamically driven activities was determined at the cortex, the time dependence of those activities was determined in the thalamus, Disregarding geometric factors, the amplitude of the cortical response is directly proportional to the number of neurons which fires its response to the stimulus. C(x, t) is equal to the number of neurons which do fire in response to the stimulus which passes through the thalamus divided by the number of neurons in the focus which could be stimulated to fire. The amplitude of the response is roughly proportional to the conditional probability of firing if stimulated. We can simplify the expression for p(x, t) to:

p(x, t ) = C(x)Th(t). Assume further that there are N intrathalamic circuits participating in the circulating activity, each of which can be in one of two states, T (transmitting) or B (blocking). At rest assume that all of the circuits are in the unexcited state, T. A stimulus passing through the thalamus at time t--0 excites all N of the circuits to state B. According to one of the most widely studied models of neuronal membrane properties, Stein's model, between stimuli membrane potential decays exponentially (Stein, 1965; Lfinsk), 1984). Let the circuit revert, therefore, from state B to state T according to a Poisson process. Let 0~> 0 be the parameter of that process. Assume further that the event that a circuit reverts occurs independently of the reversion of any other circuit. Suppose that a second stimulus arrives at time t after the first and that it is transmitted to the cortex if and only if more than m thalamic reverberating circuits have reverted to state T. Let g(i,t)=[exp(-~t)][aiti]/i!. The probability that the arriving stimulus is transmitted through the thalamus is given by: N

Th(t)=K

~

g(i,t),

i--m+ l

where K > 0 is a normalization constant. The model bears some resemblance to a thermodynamic model of thalamocortical responses in which the cortical response is a product of a non-time dependent cortical factor and a sum of time dependent thalamic factors (Chan and Wing, 1978). Classically the epileptic property of a focus is thought to reflect hyperexcitability of the neuronal aggregate (Engel, 1987). Stimulation of the ventrolateral

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thalamic nucleus, eliciting the thalamocortical augmenting response, makes it easier to produce cortical epileptiform discharges (Steriade and Yossif, 1974). The occurrence of cortical epileptic activity is associated with enhancement of the thalamocortical augmenting and recruiting responses (Kostopoulos, 1982). Engel (1987) has emphasized the increased synchronization of the neurons underlying the epileptic activity. Enhanced inhibitory mechanisms could give rise to pronounced neuronal synchronization of the type associated with the thalamocortical augmenting and recruiting responses (Engel, 1987). During the time that the cortical neurons are freed from the effect of the input passing through the thalamus, which blocks any input while sufficient neurons are in state B, the cortical neurons are free to generate the self sustained activity which manifests itself as an epileptic seizure. Thus the period during which a sufficient number of thalamic circuits are in state B would seem to be the period when epileptic activity could be generated in the cortical neurons. Kokkinidis and Anninos describe GSW in their model of two interacting neuronal populations (1985). The thalamocortical systems underlying spindles, recruiting responses and GSW are thought to be closely related (Gloor, 1984). Gloor and Fariello (1988) postulate that some of the thalamocortical neurons involved in spindle mechanisms are involved in feline generalized penicillin epilepsy. Jasper (1949) discusses the thalamic reticular system, in which he includes both the midline-intralaminar thalamic nuclei and the nucleus reticularis thalami (RE), as controlling cortical rhythmic activity and specifically GSW. The mechanisms underlying thalamic rhythmic activity are probably also involved in GSW. Epileptiform activity induced by cortical application of penicillin inhibits transmission through thalamic relay nuclei (Ogden, 1960). Elements of the thalamic sequences of excitatory and inhibitory postsynaptic potentials which underlie thalamocortical rhythmic activity have been observed during the induction of cortical epileptiform activity by penicillin (Steriade and Deschines, 1984). Corticothalamic stimulation is the most effective way of eliciting these sequences (Steriade and Desch~nes, 1984). Skondras obtained neural activity resembling GSW by appropriately altering the initial parameters of his model of a single network of undifferentiated neurons (1988). Low threshold calcium spikes (LTS) have been identified in only a small fraction (about 10%) of neocortical cells but are present in almost all thalamocortical neurons (Steriade and Llinhs, 1988). LTS are elicitable in a neuron only after it has been hyperpolarized (Prince, 1988). Although other physiological phenomena intrinsic to the thalamic neuron, such as the persistent depolarizing sodium current, the voltage dependent potassium currents, other calcium currents and the afterspike hyperpolarization may participate in thalamic rhythmic activities, LTS appears to play the primary role (Steriade and Llin~s, 1988). Ethosuximide and trimethadione, two

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anticonvulsants effective against a type of epilepsy characterized by GSW, inactivate a low threshold calcium current in thalamic neurons. The greater the hyperpolarization of the thalamic neurons, the greater the ethosuximide induced inactivation (Coulter et al., 1989). The effects upon LTS suggest that the initial excitation and subsequent inhibition postulated by the model as underlying epileptiform activity do occur. The model differs from previous models of epilepsy in that the effect of external stimuli is thought of as essentially inhibitory rather than excitatory or mixed excitatory and inhibitory. Blocking perturbing external input might allow the synaptic connections to become stronger. Thus the model may bear some relation to the concept of Anninos and Cyrulnik (1977) and of Kokkinidis and Anninos (1984) that increased strength of cortical connections may be related to epileptogenesis. Blocking cortical inhibition could therefore lead to the increase in the probability of interconnection which is reflected in synchronous neuronal bursting (Traub and Wong, 1983). In addition, blocking input to the cortex would increase the variability in the spontaneous activity of the neurons in the cortical netlet. If that spontaneous activity exceeds a critical value, epileptic activity would ensue (Kokkinidis and Anninos, 1984). 3. Intrathalamic Circuits. Verzeano and Negishi (1960) found circulating activity in the thalamus elicited by peripheral stimulation, suggesting that such intrathalamic circuits do exist. Other than their location in the thalamus, no assumption has been made regarding the anatomical identity of the intrathalamic circuits. A precise identification of the circuits is not currently possible on the basis of available experimental data. Because of their importance in the model, it is warranted to identify the circuits as fully as possible. At this time it is unclear whether the postulated intrathalamic circuits are part of the reciprocal connections between RE and thalamocortical relay cells, involve the thalamic local circuit neurons interposed between thalamocortical relay cell and RE or consist of other components .The differential roles of RE and thalamic local circuit GABAergic cells remain to be defined (Steriade and Llinfis, 1988). The possible contribution of the intralaminar nuclei also awaits further clarification (Macchi and Bentivoglio, 1986). Reciprocal connections between medial and lateral thalamus have been demonstrated electrophysiologically (Desiraju et al., 1969). GSW is thought to be a manifestation of dysregulation in thalamic systems including the intralaminar nuclei (Macchi and Bentivoglio, 1986). Purpura felt that the inhibitory effect of medial thalamic stimulation on thalamocortical neurons participated in preventing non-relevant stimuli from reaching the cortex (1970). This inhibitory sculpting of afferent impulses is akin to the function of the thalamus in the tonically active (relay) mode, which Sterriade and Llin~s

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(1988) clearly distinguish from the thalamic oscillatory mode of functioning, which underlies thalamic rhythmic activities. Although one cannot be ruled out, the weight of current evidence appears to be against the intralaminar nuclei having a primary role in GSW. Most thalamocortical axons give collaterals to RE on their way to the cortex (Steriade and Llin~s, 1988). Those collaterals are excitatory (Steriade et al., 1984). Many investigators agree that RE projects reciprocally to the thalamic nuclei of origin of the thalamocortical axons (Jones, 1984; Scheibel, 1984; Steriade and Desch~nes, 1984; Herkenham, 1986; Steriade and Llinfis, 1988). Based on the GABAergic nature of those projections, the effect of RE activity on thalamocortical relay neurons is generally presumed to be inhibitory (Steriade and Llin~s, 1988). RE neurons also synapse on GABAergic thalamic local circuit neurons. Activity transmitted through that circuit would inhibit the inhibitory local circuit neurons with resultant excitatory effect on the thalamocortical relay cells (Steriade and Llin~s, 1988). Thus it is also possible that RE might exert excitatory actions on thalamocortical relay neurons (Steriade et al., 1984). Although they feel that the local circuit neurons participate primarily in the discrimination of stimuli, Steriade and Llinfis (1988) speculate that the differing effects of RE activity on thalamocortical relay neurons depending upon how that activity is transmitted might explain the well known effect of sleep in altering the likelihood of seeing GSW discharges in epileptic patients. Anatomic connections between the pacemaker of thalamic rhythmic activity and the thalamocortical relay nuclei, the final path for the transmission of the rhythmic activity to the cortex, are necessary for the thalamic rhythmic activity to become generalized (Steriade and Llin/ts, 1988). This resembles Kusske's (1976) postulated mechanism for the spread of focal epileptiform activity to become generalized. The evidence for connections between the intralaminar and relay nuclei of the thalamus derives primarily from Golgi and anterograde deterioration studies and is largely unsupported by evidence based on axonal tracing (Macchi and Bentivoglio, 1986). Recent retrograde tracing studies with fluorescent tracers have, however, provided some evidence for the existence of such connections (Steriade and Desch6nes, 1984; Macchi and Bentivoglio, 1986). Steriade and Desch6nes caution that uptake by passing fibers cannot be ruled out (1984). Presumably based on this paucity of support, Jones has concluded that the intralaminar nuclei connect only with other intralaminar nuclei and not with other portions of the thalamus (1984). RE, however, does have the widespread thalamic projections needed to induce r h y t h m i c hyperpolarization-rebound requences in almost all relay nuclei (Steriade and Llinfis, 1988). Pathways involving RE are important components in the propagation of epileptiform activity, in the development of spindles limited to a local area of

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cortex into generalized spindles with the associated clinical manifestations (Steriade and Llin~ts, 1988). RE is involved in the genesis of the rhythmic thalamocortical spindle activity (Jones, 1984; Scheibel, 1984; Steriade and Desch6nes, 1984). It has been proposed that the GABAergic RE cells induce the spindle rhythm and evidence in support of that hypothesis has been obtained (Steriade and Llin&s, 1988). RE axons preferentially terminate in the thalamic intralaminar nuclei and the intralaminar nuclei project reciprocally to RE (Steriade, Parent and Hada, 1984). Such interconnections might explain the effect of intralaminar stimulation on thalamic rhythmic activities within the context of the hypothesis of RE as the pacemaker of those activities. The generally accepted position assigns to RE a principal role, in some form, in the genesis of thalamic rhythmic activities and, by implication, in GSW (Scheibel, 1984; Steriade and Desch6nes, 1984; Steriade and Llin~ts, 1988). 4. Results 4.1. Developmental correlations.

It can be shown that: N

[g(i, t)] [(i/t)--o~].

Op(x, t)/Ot = KC(x) i-m+

l

Since C(x) is a probability and the case where C(x) = 0 is physiologically trivial, C(x)>~0. g(i, t)~>0 for all i and t~>0, the biologically realistic cases. When t is large, Op/Ot < 0, that is the effect of the thalamocortical circuitry on cortical activity is inhibitory. Anticonvulsants such as ethosuximide which are effective against absence attacks are effective at large values of t (Nowack et al., 1979). Fromm et al. (1980; 1986) have shown that anti-absence drugs are effective against central inhibition. Similarly, when t is small, the effect of the thalamocortical circuits is excitatory. In view of the observation that activity in the intrathalamic circuits inhibits thalamocortical cells, this temporal sequence of activity is similar to that induced in thalamocortical relay cells by a cortical penicillin induced epileptic focus (Gutnick and Prince, 1975) and recalls the LTS in previously hyperpolarized thalamocortical cells (Prince, 1988). This sequence is also reminiscent of the cyclical inhibition and excitation which occurs in thalamocortical cells during experimental GSW (Avoli et al., 1983). The effect of an anticonvulsant drug can be measured by looking at the ratio R ofp(x, t) after to p(x, t) before administration of the anticonvulsant. Since the summation in the expression for p(x, t) is a portion of a convergent series (in the first approximation the terminal portion) and m is large, where Kis a constant and A~ is the change in ~, we may approximate R by K exp(-- A~t). The curve of R vs t for ethosuximide resembles that which would be generated only if - - A ~ < 0 (Nowack et al., 1979). Since 1/~ is the relaxation time of an intrathalamic circuit, ethosuximide causes the intrathalamic circuit to remain

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in the excited state for a shorter time before relaxing to the non-excited state. Ethosuximide also attenuates inhibition (Fromm et al., 1980; F r o m m , 1986). Thus the inhibitory nature of exciting the intrathalamic circuit is consistent with the model. Let t* be the value of t for which Th(t) is a maximum. Consider Th(t*) as a function of e and t* on the population of subjects. The expressions for the partial derivatives of Th(t*), may be calculated from Th(t) and d T h ( t * ) = 0 . Differentiating implicitly it can be shown that dt*/de = - (t*/oO. Since • > 0 and t* >0, dt*/dc~0. Symmetrically extend R to the cases which never occur. For therapeutic doses of phenytoin d R / d t - O (Nowack et al., 1979). Since phenytoin in nontoxic doses does not markedly affect the evoked response to a single stimulus (Kaplan, 1977), C2(x )-- C 1(x). The denominator of dR/dt is nonzero because the alternative is not compatible with a living organism. The numerator of dR~dr is a polynomial in t which is equal to 0 for all values of t. Since the coefficient of the highest order term equals the product of a nonzero term times ( e l - e 2 ) , it follows that 7~=e2, for physiologic doses of phenytoin. Since epileptiform discharges can be seen in a single cortical neuron (Steriade and Yossif, 1974), to a first approximation m = N. Phenytoin exerts its antiepileptic effect by altering N. The event that a second peripheral stimulus occurring at time t after the first stimulus in the pair is not transmitted through the thalamus is equivalent to the event that the waiting time until a sufficient number of intrathalamic circuits

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relaxes from state B to state T is greater than t. The waiting time until the occurrence of a fixed number of independent events, each of which occurs according to identical Poisson processes, is distributed according to a gamma distribution with integral shape parameter (Parzen, 1962). Where (IV> is the expected value of W, the waiting time, Hastings and Peacock (1974) have shown for such a distribution, sometimes known as an Erlang distribution, that: = (m+ 1)/at where e and m are defined as above. Since each intrathalamic circuit reverts to state T according to a Poisson process with parameter a, the expected waiting time until that relaxation is given by 1/e. The duration of an excitatory postsynaptic potential in a single thalamic neuron induced by stimulation of the medial thalamus is approximately 60-80 msec (Purpura and Cohen 1962). Verzeano and Negishi (1960) found three neuron circulating activity in the thalamus. It would therefore be reasonable to estimate that 1/e should average 210 msec. Schallek and Kuehn (1963) found that the average duration of an epileptiform afterdischarge in a cat thalamic neuron, (W>, was 51333 msec. The number of intrathalamic cells excited by an incoming stimulus is approximately 245. Estimates of the number of neurons in the human cortex have ranged from 2.6 billion to 50 billion (Pakkenberg, 1966; Angevine and Cotman, 1981). The surface area of the h u m a n cortex has been estimated at from 2200 cm / to upwards of 4000 cm 2 (Angevine and Cotman, 1981; Carpenter and Sutin, 1983). Bok (1959) studied different species of rodents and concluded that there were approximately 1530 cortical neurons underlying a pial area of 0.01 m m z. He also concluded that this value is essentially constant in mammalian species. Epileptic activity is a property of an aggregate of interconnected neurons (Scobey and Gabor, 1977). Anatomical-physiological analysis has shown that paroxysmal activity depends upon small, vertically oriented columnar organizations of neurons. Minute spike generators have been observed in epileptogenic cortex (Goldensohn, 1985). A single feline cortical generator zone, which is adequate to generate synchronized cortical activity and possibly also ictal activity, occupies a cortical area 0.12 m m in diameter (Gabor et al., 1979). Thus there are at least 1700 neurons in the minimal cortical area which can support epileptic activity.-It has proven difficult to specify how many neurons are needed to support epileptic activity and, using other experimental techniques, other investigators have found different, usually larger, upper limits for the minimal size of epileptic cortex (Reichenthal and Hocherman, 1979; Goldensohn, 1985). Depending on the species and system studied, the number of cortical neurons

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attached to a subcortical relay cell varied widely from about 25 to 200 (Blinkov and Glezer, 1969). In more recent work, Creutzfeldt (1978) has found that the early measurements were overestimates and that many thalamic neurons are associated with only one or at most few cortical neurons. A conservative estimate for the number of cortical cells stimulated by a single thalamic cell would therefore seem to be 20. In dealing with the circuitry of the cortex, both direct and indirect input pathways from the thalamus must be considered (Jones, 1988). The innervation of small cortical areas by many thalamic cells and the divergent projection from individual thalamic nuclei to more than one cortical area suggests that anatomical reality may be more complex than suggested in the conceptual model (Goldman-Rakic, 1988). The excitability of a neocortical neuron is a result of the interplay of multiple biological factors, many of which may vary with time (Prince, 1988). A similarly complex array of influences plays on the thalamocortical neuron (Steriade and Llin/Ls, 1988). Recognizing that a conceptual model involves some degree of simplification of reality and that, therefore, the model might not present a picture accurate in every detail at each instant of time, we assume that each thalamocortical relay cell participates in just one thalamic circuit. Although experimentally unproven as of yet, the assumption is reasonable. Eighty-five thalamic circuits underlie a minimal epileptogenic cortical area. A therapeutic anticonvulsant dose ofphenytoin, in a species and with a route of administration similar to that used by Schallek and Kuehn, is 15 mg/kg (Louis et al., 1968). A dose of 10 mg/kg of phenytoin, which would not be expected to have anticonvulsant effect, reduced the duration of the epileptiform afterdischarge (W) by only 7000 msec (Schallek and Kuehn,1963) or N by about 33 circuits. Thus an incoming stimulus would excite about 212 thalamic circuits, thereby freeing more than the minimum number of cortical neurons needed to support epileptiform activity. This finding is consistent with the expected lack of anticonvulsant effect of this lower dose. Intravenous administration of 20 mg/kg ofphenytoin decreases Wby 33700 msec (Schallek and Kuehn, 1963). Since the calculated change in Nis about 160, it follows that, to this first approximation, an incoming stimulus excites only 85 thalamic circuits. This implies that, following an anticonvulsant dose of phenytoin, an incoming stimulus no longer excites enough neurons to sustain epileptic activity. Phenytoin can exert an anticonvulsant effect, at least primarily, by only altering N. Trimethadione, although since replaced in clinical use by other medications, is an example of a drug which is effective against non-convulsive but not against convulsive seizures. Administration of 400 mg/kg of trimethadione, which is within the reported range of therapeutically effective doses (Everett and Richards, 1944; T o m a n et al., 1946), reduces W by 29800 msec (Schallek and Kuehn, 1963). With the change in N calculated in the same way as for

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phenytoin, an incoming stimulus would excite over 100 intrathalamic circuits, more than enough to allow epileptic activity. This seems to contradict the antiepileptic effect of trimethadione. The difference between the effects of phenytoin and trimethadione suggests that the assumption, that only N is changed by trimethadione, may not be correct. Because of its toxicity, trimethadione has been replaced by a chemically unrelated medication, ethosuximide, for the treatment of petit real absence seizures. As shown for trimethadione, it is likely that treatment with ethosuximide leaves N at a sufficiently large value for the approximations made below. Ethosuximide does not alter the response to a single stimulus ((~apek and Esplin, 1977). Therefore, it is reasonable to assume that C(c) remains unchanged by ethosuximide treatment. We therefore have for ethosuximide:

.i2,z/i,)If[K.exp(--.,t)(i=.~÷1 .,fl/i,)].

R= IK2exp(--o%0 (,=.~+'

N is large and the sums in the numerator and denominator may be approximated with the terminal parts of infinite series. Since both series are convergent: R=-K 3

exp((~ 1 - ~2)t),

where K 3 is a constant. Thus the effect of ethosuximide is primarily upon a. 5. Discussion. Antiepileptic medications often have multiple biochemical and neurophysiological effects. Current technology often does not allow the investigator to be certain which of the changes caused by a drug are the basis for its antiepileptic effect (McDonald, 1983). For example, the multiple effects of phenytoin have been extensively studied, but which effects are related to phenytoin's anticonvulsant effect is unclear (Morselli and Lloyd, 1985). The mechanism of action of ethosuximide is also unknown (Morselli and Lloyd, 1985). With the exception of valproic acid, antiepileptic drugs are effective against convulsive or non-convulsive seizures but not both. F r o m m (1987a) has speculated that one of the sources of confusion in the literature regarding epilepsy has been failure to distinguish between effects on convulsive and nonconvulsive seizures. Changes in thalamocortical excitability can explain the different effects of anticonvulsant drugs in feline models of primary generalized epilepsy and of secondarily generalized temporal lobe epilepsy (Shouse, 1987). Nowack et al. (1979) have shown differential changes in thalamocortical excitability which correlate with those different profiles of clinical antiepileptic efficacy. Phenytoin, a typical drug effective against convulsive seizures, appears to exert its antiepileptic effect primarily by altering N. Ethosuximide and

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trimethadione, typical drugs effective against non-convulsive seizures, exert their antiepileptic effects primarily by altering e, although it is possible they may also alter N. Gloor (1979) observed that in occasional cases Type I epileptogenesis can lead to Type II epileptogenesis and convulsive seizures and concluded that the two types of epileptogenesis represent extremes ends of a continuous spectrum of epileptic conditions. Fromm (1987a) feels that the two types of epileptogenesis are qualitatively and clinically different phenomena which ought to be considered separately. The transition to cortical activity resembling that seen during convulsive seizures is brought on by a breakdown in cortical recurrent inhibition (Kostopoulos et al., 1983). Such a breakdown has also been noted in models of partial and convulsive seizures (Gloor, 1989). Whether this differential preservation of recurrent cortical inhibition represents two completely different states (Gloor and Fariello, 1988; Gloor, 1989), or just a gradual change along a; continuum (Gloor, 1979) is unclear. Epstein and Andriola (1983) have speculated that the two types of epileptogenesis are extremes which rarely occur in pure form and that most clinical seizures represent a mixture of the two types of epileptogenesis. Deductions from the model of thalamocortical excitability and interactions support Fromm's position that convulsive and non-convulsive seizures are qualitatively different phenomena. Wilson and Cowan (1973) model the brain, specifically the cortex and the thalamus, as consisting of multiple two-dimensional layers of neurons interconnected within their layer. Most of the current neurophysiological epilepsy research has focused on the changes in individual cortical neurons, the changes within neurons in a single layer, which lead to seizures (Jeffreys and Roberts, 1987). Several years ago Jasper noted that the phenomenon of epilepsy will not be understood only by studying the properties of individual neurons (1969). The thalamocortical model extends that observation and demonstrates the importance of also taking into consideration the role of interactions between cortical and subcortical structures in the generation of generalized seizures. Muir has discussed the balance between the introduction of simplifying assumptions to make the mathematics of a neural network model tractable and the preservation of the biological reality of a model (1981). In practice theoretical models have tended to be limited to a population of randomly connected homogeneous neurons in order to simplify the computations. Because of these assumptions, neural models have yielded some useful results but generally have been of limited applicability and are often limited to the specific situation which they model (Harmon and Lewis, 1966). The practical role of theoretical models of epilepsy thus remains to be defined (Kaczmarek and Babloyantz, 1977). The thalamocortical model demonstrates that the

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i n c l u s i o n of a small a m o u n t of biological v e r s i m i l i t u d e , limited b y e x t a n t r e l e v a n t e x p e r i m e n t a l d a t a a n d the desirability of m a i n t a i n i n g m a t h e m a t i c a l t r a c t a b i l i t y the m o d e l , c a n lead to c o n c l u s i o n s n o t a l r e a d y explicit in the b i o l o g i c a l d a t a u p o n w h i c h the m o d e l is based. T h e p o w e r of a c o m b i n a t i o n of m o d e r n s i m u l a t i o n t e c h n o l o g y a n d a limited a m o u n t of biological realism h a s b e e n e l e g a n t l y d e m o n s t r a t e d (Miles et al., 1988). It is n o t u n r e a s o n a b l e to expect t h a t f u r t h e r d e v e l o p m e n t of a n a l y t i c a l a n d c o m p u t a t i o n a l t e c h n i q u e s will e n a b l e the i n c l u s i o n of g r e a t e r a m o u n t s of b i o l o g i c a l d a t a in m o d e l s , e n a b l e steadily s t r o n g e r c o n c l u s i o n s to be d r a w n f r o m m o d e l s a n d thus increase the utility of t h e o r e t i c a l m o d e l i n g to b i o l o g y a n d m e d i c i n e .

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