DYNAMICS OF NON-CONVULSIVE EPILEPTIC PHENOMENA MODELED BY A BISTABLE NEURONAL NETWORK

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1 Neuroscience 126 (24) DYNAMICS OF NON-CONVULSIVE EPILEPTIC PHENOMENA MODELED BY A BISTABLE NEURONAL NETWORK P. SUFFCZYNSKI, a,b * S. KALITZIN a AND F. H. LOPES DA SILVA a,c a Stichting Epilepsie Instellingen Nederland, Achterweg 5, 213 SW Heemstede, The Netherlands b Laboratory of Medical Physics, Institute of Experimental Physics, Warsaw University, Hoza 69, -681 Warsaw, Poland c Section Neurobiology, Swammerdam Institute for Life Sciences, University of Amsterdam, Kruislaan 32, 198 SM, Amsterdam, The Netherlands Abstract It is currently believed that the mechanisms underlying spindle oscillations are related to those that generate spike and wave (SW) discharges. The mechanisms of transition between these two types of activity, however, are not well understood. In order to provide more insight into the dynamics of the neuronal networks leading to seizure generation in a rat experimental model of absence epilepsy we developed a computational model of thalamo-cortical circuits based on relevant (patho)physiological data. The model is constructed at the macroscopic level since this approach allows to investigate dynamical properties of the system and the role played by different mechanisms in the process of seizure generation, both at short and long time scales. The main results are the following: (i) SW discharges represent dynamical bifurcations that occur in a bistable neuronal network; (ii) the durations of paroxysmal and normal epochs have exponential distributions, indicating that transitions between these two stable states occur randomly over time with constant probabilities; (iii) the probabilistic nature of the onset of paroxysmal activity implies that it is not possible to predict its occurrence; (iv) the bistable nature of the dynamical system allows that an ictal state may be aborted by a single counter-stimulus. 24 IBRO. Published by Elsevier Ltd. All rights reserved. Key words: absence, epilepsy, seizures, thalamocortical. *Correspondence to: P. Suffczynski, Medical Physics Department, Stichting Epilepsie Instellingen Nederland, Postbus 54, 213 AM Hoofddorp, The Netherlands. Tel: ; fax: address: psuffa@sein.nl (P. Suffczynski). Abbreviations: AMPA, -amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid; DC, direct current; EEG, electroencephalogram; GAERS, genetic absence epilepsy rat from Strasbourg; IN, interneuron; IPSP, inhibitory postsynaptic potential; LTS, low threshold spike; NMDA, N-methyl-D-aspartate; pps, pulses per second; PY, pyramidal; RE, reticular; SW, spike and wave; TC, thalamocortical; WAG/Rij, Wistar albino Glaxo from Rijswijk /4$ IBRO. Published by Elsevier Ltd. All rights reserved. doi:1.116/j.neuroscience Absence seizures are paroxysmal losses of consciousness that start and end abruptly and are accompanied by bilaterally synchronous spike and wave (SW) discharges that can be recorded on the electroencephalogram (EEG). The cause of this abnormal behavior is presumably genetic although genetic abnormalities responsible for absence phenotype in humans are not yet clear (Crunelli and Leresche, 22a). The development of animal models of absence seizures and, more recently, computational models of circuits involved in SW generation, advanced our understanding of the basic neuronal mechanisms of this type of epilepsy. It is currently believed that the mechanisms underlying SW discharge may be related to the thalamocortical mechanisms of sleep spindle generation (Steriade et al., 1993; Avoli et al., 21; McCormick and Contreras, 21). However the mechanisms that are responsible for the spontaneous transition between the normal spindle oscillations and paroxysmal SW activity are not well understood. Similarly, the questions concerning the factors that determine seizure duration and/or inter-seizure interval have also not been often addressed. Experimental studies in animal models have yielded new findings (reviewed in McCormick, 22; Destexhe and Sejnowski, 23) but these are still fragmentary such that a comprehensive interpretation of the role played by different mechanisms in the process of seizure generation, both at short and long time scales, is still limited. We assumed that a computational model that takes into account the most relevant (patho)physiological experimental findings would be an appropriate tool that could contribute to reach such a comprehensive view. A number of detailed, distributed models of thalamic and thalamo-cortical networks were recently developed (e.g. Wang et al., 1995; Lytton et al., 1997; Destexhe 1998, 1999). These models have given insight into some basic neuronal mechanisms of SW discharges, but do not address specifically the most essential issue of this type of epileptic activity: that a given thalamocortical loop can display both kinds of activity without specific adjustments of parameters being expressly made. Indeed the essence of epilepsy is that a patient displays (long) periods of normal EEG activity (i.e. non-epileptiform) intermingled with epileptiform paroxysmal activity only occasionally. Thus, the main aim of the present study is to find out the mechanisms responsible for transitions from normal activity to paroxysmal SW discharges. This aspect of the dynamical process responsible for epilepsy has, to the best of our knowledge, not been addressed in a computational model of absence epilepsy and therefore the present model provides a novel contribution. In this study we approach the given problem at the intermediate level between the distributed neuronal network and lumped circuit levels. That is, we do not simulate the explicit behavior of individual neurons but rather model the populations of interacting neurons integrating neuronal and network properties. While the correspondence between distributed neuronal networks and population models has been established both theoretically (Wilson and

2 468 P. Suffczynski et al. / Neuroscience 126 (24) Cowan, 1972) and experimentally (Freeman, 1975, 1979) the advantages of the lumped approach over more complex models are, at least, three-fold. First, the relatively simple model which is able to replicate specific experimental results enables one to better understand which real system s properties are necessary and sufficient to account for the particular observed phenomena. Second, such approach enables to investigate system dynamics at the macroscopic level, that is at the level where electric brain signals such as local field potentials or EEG are recorded. Overall system dynamics is usually hardly accessible in distributed neuronal models. Third, due to high computational efficacy, we are able to simulate long time range (hours or days) behavior and in this way we can investigate in a realistic time scale the durations of intervals between seizures. In the present study we focus on the SW activity that is commonly recorded from a genetic animal model of absence epilepsy, namely the Wistar albino Glaxo from Rijswijk (WAG/Rij) rat (van Luijtelaar and Coenen, 1986; Coenen and van Luijtelaar, 23), that has similar characteristics to the generalized absence epilepsy rat from Strasbourg (GAERS) model (Marescaux et al., 1992; Danober et al., 1998). The main reason to choose these animal data for constructing the computational model is that rat experimental data at the cellular and network levels are available, in contrast to the human case. Model s structure EXPERIMENTAL PROCEDURES Fig. 1. Schematic structure of connections in the thalamo-cortical network model consisting of cortical and thalamic modules. The cortical module consists of two interconnected populations of PY and IN neurons. The thalamic module consists of two interconnected populations of the TC and RE neurons. TC cells project to both the PY and IN cells, while PY cells project to both the TC and RE cells. The PY population receives external cortical excitatory input, the TC population receives external sensory input and the RE population receives external inhibitory input. Figure s legend shows synaptic connection types. The present model is an extended version of the lumped model initially proposed by Lopes da Silva et al. (1974). The latter consisted of a single module, i.e. a cluster of neurons containing two interacting populations. Excitatory cells of main population projected to the interneurons (IN) while the latter fed back on the main cells, inhibiting them with fast GABA A receptor-mediated inhibitory postsynaptic potentials (IPSPs). Each population was described by the time courses of postsynaptic potentials and sigmoid transfer function, which was used to simulate the conversion between the mean membrane potential of a neuronal population and the firing rate (number of pulses per second [pps]) of the population. We made a number of extensions and added new features to the above-described model. The present model is advanced with respect to its precursor in three main points. (i) It consists of two modules, cortical and thalamic ones, that are mutually interconnected. (ii) The transformation between mean membrane potential and firing density in the thalamic populations takes into account the low-threshold I T calcium current that underlies burst firing in the thalamic cells. (iii) In addition both fast GABA A and slow GABA B receptor-mediated inhibitions are included and GABA B receptors are assumed to have nonlinear activation properties. The schematic diagram of the model is shown in Fig. 1. The thalamic loop consists of a population of thalamocortical (TC) cells that projects to a population of reticular thalamic (RE) cells. The latter inhibits TC population by way of GABA A and GABA B types of inhibition. The TC cells receive external excitatory input that represents sensory inputs from the ascending afferents while the RE population receives external inhibitory input. The latter represents the input from the neighboring RE cells, since these are interconnected by mutual inhibitory synapses (Sanchez-Vives et al., 1997). The cortical module consists of a negative feedback loop formed by interacting populations of pyramidal cells (PY) and inhibitory IN. PY cells, in addition to projecting to IN, send also excitatory connections to the thalamus both to the TC and RE populations. In turn, the TC cells excite both the PY cells and IN. The PY population receives also external cortical input that stands for the glutamatergic input from other PY cells, not included in the lump. All excitatory synaptic interactions are mediated by glutamate -amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) receptors. The output of the model, is the mean membrane potential of the PY cell population. We assume that the dynamics of this activity (e.g. dominant frequency) is reflected on the local field potential recordings and EEG signals that have the same dynamical properties. However, in the present model we did not aim to simulate realistic field potential waveforms. A detailed model description is given in Appendix. The present modeling work was carried out using the Simulink toolbox in Matlab (The MathWorks Inc., Natick, MA, USA). One minute of simulated time took 14 s to run on the 1 MHz Pentium III PC, 512 MB RAM. All further postprocessing analysis was done in Matlab. Statistical methods The distributions of the durations of paroxysmal and paroxysmalfree epochs (Figs. 4, 5) were fitted with exponential functions using expfit procedure in Statistical Toolbox in Matlab. The distributions means and the 95% confidence intervals of the means were obtained from the fit. Subsequently we applied the onesample Kolmogorov-Smirnov goodness of fit test (more powerful than its alternative, the 2 test). In this way we tested whether the observed scores in the sample can be considered to belong to a population having a given theoretical distribution. The critical values of test statistics were taken from Siegel (1956). Cumulative

3 P. Suffczynski et al. / Neuroscience 126 (24) Fig. 2. Example of model output. Upper panel: 2 s of a simulation with the occurrence of a spontaneous paroxysmal episode. Lower panel: power spectra of simulated normal and paroxysmal activity. Dominant frequency of normal activity is around 11 Hz while that of paroxysmal activity is around 9 Hz. histograms on logarithmic scale (Fig. 4, insets) were fitted with linear functions using polyfit procedure in Matlab. RESULTS Different types of model behavior The model may exhibit two qualitatively different types of behavior, such as seen in the experimental animals (WAG/ Rij and GAERS). The output signal may display a waxing and waning spindle-like oscillation having a spectrum with a peak at approximately 11 Hz or a high amplitude seizure-like oscillation at a frequency around 9 Hz. We refer to the former behavior as normal on-going activity while to the latter as paroxysmal activity. For the reference set of parameters, the model is in a bistable mode where it may generate both normal and paroxysmal oscillations and spontaneous transitions between these two types of behavior. The model output in this mode is shown in Fig. 2, upper panel. The lower panel depicts the corresponding power spectra. Spindle activity in the model is generated by the cyclical interaction between the TC and RE populations. IPSPs of RE origin facilitate the generation of rebound low threshold spikes in TC population, which in turn activate GABAergic RE neurons. The dominant frequency (approximately 11 Hz) of this rhythmic activity is largely determined by the time courses of both AMPA and GABA A postsynaptic currents and membrane time constants and to a lesser degree by other factors like the characteristics of low threshold spikes (LTS), synaptic coupling constants and sensory input level (see Appendix). Thalamic-generated spindle activity induces activity at the same frequency in the cortical populations. Normal activity in the model consists of continuous spindles of variable, waxing and waning amplitude but without clear periodicity of occurrence, while in the experimental animals the spindles occur as well-separated and recurring sequences. Modeling of the latter pattern would require additional neuronal mechanisms that were not implemented here since modeling of realistic spindle characteristics was outside the scope of this study (see Discussion). During normal activity, the thresholds for the activation of GABA B receptors in the TC and PY cells are not reached. When one of these thresholds is surpassed, GABA B receptor mediated IPSPs may be triggered, which results in the sudden transition of the model s behavior from normal on-going activity to seizure-like 9 Hz large amplitude oscillations. The frequency and amplitude of paroxysmal oscillations depend on the relative contribution of GABA A and GABA B components. A reduction of the contribution of GABA A and an increase of that of GABA B slows down the paroxysmal oscillations and increases their amplitude. A complete block of GABA A receptors in the TC and PY cells (while keeping the GABA B conductance at the reference level) slows the oscillation down to 4 Hz. The dynamical mechanisms of spontaneous transitions between normal and paroxysmal behavior are described below. Bifurcation analysis First, the model was run with constant input signals, i.e. with the variance of noise of cortical and sensory inputs (see Appendix) equal to zero. Under such conditions the model exhibited either a constant output, or a periodic 9 Hz oscillation, depending on the initial conditions, but it did not exhibit transitions from one state to another. It indicates that transitions in the model behavior, such as shown in Fig. 2, cannot take place when input signals are constant. It suggests that random fluctuations that are introduced into the system by noise in cortical and sensory inputs are necessary for the transitions to occur. This became clearer while analyzing bifurcation diagrams. In Fig. 3A we show the two-dimensional bifurcation diagram for the reference set of parameters, as a function of constant (noise free) cortical input P Cx. Its threedimensional version is presented in Fig. 3B. In the bifurcation analysis the noise component was removed from

4 47 P. Suffczynski et al. / Neuroscience 126 (24) the inputs to keep the illustration simple. The bifurcation diagram was constructed by first increasing and subsequently decreasing the values of P Cx, the arrows along the diagram follow changes of system s output as the cortical input increases and decreases. The changes in P Cx were made step-wise and allowed the system to reach its stationary behavior. Therefore the diagram depicts dynamics of the system corresponding to different (fixed) values of the control parameter. Fig. 3A reveals that there are two types of activity possible: an equilibrium point corresponding to normal on-going activity (solid line) and a limit cycle corresponding to paroxysmal oscillations (open circles mark their minimal and maximal amplitude). Both equilibrium point and a limit cycle are emergent property of our model for the reference set of parameters that were chosen according to available physiological data. Fig. 3A reveals further that the transition between the normal ongoing and the paroxysmal activity modes occurs when the cortical input P Cx increases beyond the bifurcation point at P bif Cx. This is a typical bifurcation of the subcritical Hopf kind (Glass and Mackey, 1988). As the cortical input is subsequently decreased there is a jump from large amplitude limit cycle oscillations back to equilibrium state but this takes place at a value of P* Cx which is less than P bif Cx. This is manifestation of hysteresis. For P* Cx P Cx P bif Cx, there are two coexisting states: normal and paroxysmal. These two dynamical states are locally stable in the sense that under small perturbations of finite-time duration the system recovers to its original state. Therefore we can call such system a bistable system. The coexistence of two attractors can be seen also in the phase space plot in Fig. 3C, which is a cross-section of the bifurcation diagram shown in Fig. 3B. In the absence of noise the normal and paroxysmal attractors correspond to equilibrium point and limit cycle, respectively. When noise is introduced to the system (both cortical and sensory inputs are superpositions of two components: a constant that constitutes the direct current (DC) offset and a Gaussian noise components representing random input fluctuations), the perturbations of the steady state give rise to small amplitude rhythmic (spindle-like) activity, while perturbations of the limit cycle result in amplitude modulation of the SW oscillations (both rhythms can be seen in Fig. 2). Spontaneous (i.e. noise induced) transitions between normal and paroxysmal state occur as follows. Let us assume that the system, initially in the normal stable state, is in the bistable domain, and that all parameters, including the DC levels of the inputs, are constant. If the (randomly fluctuating) value of P Cx becomes greater than P bif Cx, the system undergoes a transition to the paroxysmal state and remains there (because this is a stable state too) until the value of P Cx becomes smaller than P* Cx, bringing the system back to its initial normal state. The dynamical scenario of transitions between normal and paroxysmal states was described above using the bifurcation diagram in Fig. 3, which was made for only one control parameter (the value of the constant component of the cortical input), and for only one system s variable (model output). A more complete bifurcation analysis that would take into account other parameters and system s variables and simultaneous variation of two or more control parameters lies beyond the scope of this paper. Additionally, noise was introduced here through the cortical and sensory inputs. One may consider, in general, that more parameters could have stochastic properties. The latter extension, however, would be of little consequence in the context of the present study (see Discussion). Therefore the following general conclusion can be formulated. The occurrence of paroxysmal activity in the model is governed by a random process, i.e. both the onset and cessation of paroxysms occur randomly over time with certain probabilities. Accordingly, the distribution of the duration of paroxysmal and normal epochs can be predicted to be exponential (Doob, 1953). Therefore the construction of histograms of the durations of paroxysmal and normal epochs from simulated and experimental time series offers a way to verify these theoretical predictions regarding the dynamics of the process leading to spontaneous paroxysmal discharges. An exponential distribution of lengths of paroxysmal and normal epochs would favor the scenario presented above; different kinds of distributions may indicate that other mechanisms have to be taken into consideration. Distributions of paroxysmal and normal epochs durations First, we constructed histograms of paroxysmal and normal epochs durations from the simulated signals. The detection of paroxysmal epochs was based on two criteria: the level of activation of GABA B receptor in the TC cells and the standard deviation of the output signal. The choice of the former criterion was based on the well-documented fact that activation of GABA B inhibition in thalamic relay nuclei is essential for paroxysmal discharges in animal models of absence epilepsy (Hosford et al., 1992; Liu et al., 1992; Snead, 1992; Puigcerver et al., 1996; Smith and Fisher, 1996; Vergnes et al., 1997; Bowery et al., 1999) although it was not confirmed in all studies (Staak and Pape, 21). In our model, the beginning of a paroxysmal epoch was detected when GABA B conductance in the TC cells was greater than 3% of the maximal conductance value and the end of a paroxysmal epoch was detected when GABA B conductance in the TC cells fell below.5% of the maximal conductance value. This choice of the detection thresholds gave very good results in comparison to visual inspection of the output signals. To improve detection performance further, we introduced a second criterion that took into account the standard deviation of the model s output amplitude. The threshold for detection of paroxysmal activity was set at three times the standard deviation of the normal, background activity. The latter was calculated on the basis of a long (1 s), paroxysmal free, signal. The epoch was considered a paroxysm when both criteria were fulfilled. The histograms of durations of paroxysmal and normal epochs detected during 24 h of simulated time, for a reference set of parameters, are shown in Fig. 4, upper panel. The histograms were fitted with expo-

5 P. Suffczynski et al. / Neuroscience 126 (24) V Cx V Cx B. 3D bif. diagram P Cx A. 2D bif. diagram * P Cx V Cx P Cx V Cx (t + τ) bif P Cx 2 2 V 2 Cx (t) 2 C. Phase portrait Fig. 3. Bifurcation diagram. (A) Two dimensional bifurcation diagram showing amplitude of the mean membrane potential of the PY cells population (V Cx in mv) along the y axis as a function of step-wise increasing and decreasing constant (noise free) cortical input (P Cx in pps) along the x axis. In the simulation performed to draw the bifurcation diagram the noise component was removed from the inputs. Arrows along the bifurcation diagram depict changes of V Cx as P Cx increases and decreases. To read the bifurcation diagram, fix the P Cx at a particular value and mentally draw a vertical line at that value. Each crossing of this line with a curve in the diagram corresponds to a point attractor (thick line) or a limit cycle (open circles denote maxima and minima of an oscillation). Point attractor corresponds to normal activity while limit cycle corresponds to paroxysmal activity. For increasing DC values of P Cx the transition from point attractor to limit cycle occurs at point P bif bif Cx. For P Cx P Cx only paroxysmal behavior is possible. For decreasing values of P Cx the transition from limit cycle to point attractor occurs at point P* Cx. For P Cx P* Cx only normal behavior is possible. For P* Cx P Cx P bif Cx the system possesses bistable dynamics: the normal state coexists with the paroxysmal oscillatory state. (B) Three dimensional bifurcation diagram of the system constructed using delay embedding with cortical input P Cx along the x axis, model s output V Cx along the y axis and delayed ( 4 ms) version of model s output V Cx along the z axis. (C) Phase portrait of the system, obtained by taking a cross-section of figure in B for P Cx 15 pps, showing coexistence of a point attractor (black dot) with a limit cycle (thin line). nential functions as described in Statistical methods. Second, we computed equivalent distributions of experimental data. These distributions were derived from data of WAG/Rij rat 6 h after administration of vigabatrin (5 mg/ kg), which significantly enhances (approximately five times) the incidence of paroxysms in these rats and therefore enables to obtain more data for analysis in a relatively short time. Data of durations of normal and paroxysmal epochs in WAG/Rij rats were obtained by an independent experimenter and details of data acquisition are given in Bouwman et al. (23). The experimental histograms and fitted exponential functions are shown in Fig. 4, lower panel. The fitted exponential functions appear to describe well both the simulated and experimental distributions. In all four examples shown, the goodness of fit test (see Statistical methods ) failed to reject the null hypothesis (H ) that the observed histogram scores came from the fitted exponential distribution even when setting a high probability,.2, of falsely rejecting H (critical values for significance levels greater than.2 were not tabulated). Additional confirmation of exponential law comes from the cumulative histograms shown in inset of each panel; in all four examples straight lines seem to fit well cumulative histograms on the logarithmic scale. Exponential distributions are characterized by a single variable, the distribution mean. In the next section the dependence of the mean duration of normal and paroxysmal epochs on model parameters is analyzed. This is particularly relevant in order to assess the influence of drug treatment on seizure frequency and/or seizure durations. Dependence of paroxysms duration on model parameters In the case that a system has two possible states, e.g. paroxysmal and normal, there are two, presumably independent, variables, that govern the system s behavior. One is the probability of the transition from normal to paroxysmal state; the other is the probability of the transition from paroxysmal back to normal state. Therefore, the resulting behavior of the system at long term can be adequately characterized by two variables. In the present study we focused on two quantifiers, namely the mean paroxysmal epoch duration and the mean normal epoch duration. Other quantifiers such as total paroxysmal duration or paroxysms incidence can be derived from these two quantifiers. The quantifiers chosen in this study have straightforward interpretation. The first one tells how long one can expect the average paroxysm to last and therefore it corresponds to the probability of transition from paroxysmal to normal state. The second quantifier, once a paroxysm has finished, tells how long we can expect to wait for a next one to occur and therefore it corresponds to the probability of transition from normal to paroxysmal state. For the analysis of the dependence of the model s behavior on parameters, we selected six of 65 model parameters. We selected the parameters that are (i) either assumed to play a role in the pathophysiology of absence seizures in animals and humans like cortical GABA A (Luhmann et al., 1995; Spreafico et al., 1993) and I T current in RE cells (Tsakiridou et al., 1995), or (ii) are assumed to be targets of antiepileptic drugs e.g. burst firing in TC cells (Coulter et al., 1989; Leresche et al., 1998) and GABA A inhibition between RE cells (Huguenard and Prince, 1994), or (iii) are associated with seizure activation procedures like sleep and hyperventilation (Niedermeyer and Lopes da Silva, 1999). We varied one parameter at a time while all others were kept constant. A multiple parameter sensitivity analysis is beyond the scope of this paper. For each parameter setting we simulated 24 h of activity and detected paroxysmal and normal epochs as described in a previous section. Duration histograms were fitted with exponential distributions and the distributions means were calculated. Each parameter (except thalamic cholinergic modulation) was manipulated such that the system s behavior varied from at least one paroxysmal event during 24 h, to a state of continuous paroxysmal activity. The influence of a cholinergic neuromodulatory input originating from the brainstem mesencephalic cholinergic neurons was investigated

6 472 P. Suffczynski et al. / Neuroscience 126 (24) No. of events No. of events 4 2 Paroxysmal epochs log cum. hist Duration (seconds) 1 5 Model simulation Normal epochs Paroxysmal epochs log cum. hist Duration (seconds) No. of events 2 Rat data No. of events log cum. hist Duration (seconds) 1 5 Normal epochs log cum. hist Duration (seconds) Fig. 4. Distributions of lengths of normal and paroxysmal epochs of the simulated signal (upper panel) and of real EEG signal recorded from the rat (lower panel). On each plot an exponential function fitted to the histogram is shown. Model histograms were obtained by simulating 24 h of activity using the reference parameters set. Histograms of rat data were obtained from 3 min recording of EEGs of WAG/Rij rat after administration of high dose of vigabatrin. In all four distributions shown, the divergence between the observed histogram and fitted function is small and we were not able to reject the null hypothesis that the observed histogram scores came from the fitted exponential distribution even when tested at high significance level (.2). Insets in each window depict cumulative histograms shown on logarithmic scale and straight lines fitted to histogram points. In all four plots the lines fit well the histograms points further confirming exponential distributions. by applying additional DC offset simultaneously to membrane potential in the TC and RE populations. This offset was varied in the range 4 to 4 mv in the TC and in the range 8 mv to 8 mv in the RE population ( 1 mv shift in TC corresponded to 2 mv shift in RE). This is justified taking into account that acetylcholine released by cholinergic pathways decreases a potassium conductance in the TC cells that brings about depolarization of the TC population, while it increases a potassium conductance in the RE neurons and thus induces hyperpolarization of the RE population (McCormick and Prince, 1986, 1987). Results of the analysis are summarized in Fig. 5. For each parameter two panels are presented where the mean duration of paroxysmal epochs (left panel) and of normal epochs (right panel) are shown. The change of a parameter is given as a percentage of the corresponding reference value. It can be seen that an increase of the duration of paroxysms and a decrease of the intervals between paroxysms can result from a series of factors: a reduction of cortical GABA A inhibition (Fig. 5A), reduction of intra-re GABA A inhibition (Fig. 5E), withdrawal of thalamic cholinergic modulation (Fig. 5F), an increase of the slope of the sigmoid in the cortical interneuronal population (Fig. 5B), an increase of burst firing in the RE or TC populations (Fig. 5C, D). The mean paroxysmal epoch duration is always a monotonic function of parameter change. In some plots this dependence is linear on a logarithmic scale (Fig. 5C, E) indicating that in these cases the mean epochs duration has exponential dependence on the varied parameter. Counter-stimulation There is a general interest to develop methods able to control the occurrence or the evolution of seizures. Thus we investigated, using the model, how a paroxysmal oscillation could be aborted. Since the model s dynamics displays bistability where a limit cycle co-exists with a steady state (Fig. 3C), it should be possible to terminate a paroxysmal oscillation by an appropriate stimulus (Glass and Mackey, 1988). We verified this theoretical prediction as follows. A paroxysmal oscillation was triggered by a pulse of 4 pps amplitude, 1 ms duration delivered to the cortical input around second 1.5. If no stimulation was applied afterward, the paroxysmal discharge continued as time progressed (Fig. 6A). If a counter-stimulation pulse with the same parameters as the trigger but shifted forward in time by 62 ms was delivered to the cortical input, the paroxysmal oscillation may be aborted (Fig. 6B). The paroxysmal oscillation, however, was annihilated only if the stimulus was delivered at a specific phase of the oscillation. Close inspection of the output signal showed that a depolarizing pulse was effective if it was delivered near the time at which the output signal reached maximal negative polarity (Fig. 6B, inset). Counter-stimulation performed equally well on spontaneous paroxysms as on those triggered by an external stimulus. Phase-amplitude analysis of the effectiveness of the counter-stimulus is shown in Fig. 6C. In simulations performed to obtain this plot, the noise component was removed from the inputs, to be sure that the termination of the paroxysmal rhythm is due to stimulation only. This rhythm was triggered as in A and B. The counter-stimulus was always a rectangular pulse of 1 ms duration, its phase was sampled from 36 in steps of 1 (assuming cycle frequency 9.28 Hz) and its height was varied from to 15 pps in steps of 1 pps. On a polar plot zero phase corresponds to maximal negative polarity of the output signal. The counterstimulation was delivered approximately 1 s after a trigger to avoid transient effects influencing the analysis. Fig. 6C

7 P. Suffczynski et al. / Neuroscience 126 (24) A. GABAa in Cx paroxysmal normal B. Slope sigmoid IN paroxysmal normal mean epoch duration (seconds) C. G RE/LTS 1 1 paroxysmal normal 1 3 E. GABAa in RE paroxysmal normal D. G TC/LTS 1 5 paroxysmal normal 1 3 F. ACh modulation paroxysmal normal parameter change (%) parameter change (%) Fig. 5. (A F) Dependence of the mean duration of normal and paroxysmal epochs on a number of model parameters (A: cortical GABA A, B: the slope of the sigmoid in the IN population, C: the amplitude of I T current in the RE population, D: the amplitude of I T current in the TC population, E: GABA A in the RE population, F: thalamic cholinergic modulation from the brain stem). For each varied parameter the mean and 95% confidence intervals of duration of paroxysmal epochs are shown on the left panel while those of normal (inter-ictal) epochs are shown on the right panel. In all graphs, the x axis denotes the relative parameter change in percentage around its reference value, while the y axis denotes the mean epoch duration, in seconds, on a logarithmic scale. shows that the abnormal rhythm can be stopped either by a positive stimulus applied on the descending slope of the PY population amplitude or by a negative stimulus applied on the rising slope of the PY population signal. Independent cortical or thalamic paroxysmal oscillations We also investigated whether bistable behavior can be present in cortical or thalamic networks separately. The results are summarized in Fig. 7. In these two diagrams we plotted the dynamics of the activities (firing densities) of the four neuronal populations (TC, RE, PY and IN) for three cases, namely the reference condition where all populations are active, and two cases where either the TC and RE or the PY and IN are inactive. The condition where TC and RE neurons are quiescent represents the activity of the cortical model, and the condition where PY and IN neurons are inactive represents the thalamic model. Each closed curve in the diagram corresponds to oscillatory (paroxysmal) activity, while the dot represents the stable (normal) state. Thus for the three different cases, normal and paroxysmal states can be shown to coexist. This means that bistability can be present in the thalamic network without input from the cortex (in Fig. 7A activity of PY cells is equal to zero on the x axis), while cortical bistability may be present without thalamic involvement (in Fig. 7B activity of TC cells on the x axis is equal to zero). In the thalamo-cortical case bistability involves all four neuronal populations as suggested by the fact that the activities of all populations contribute to the corresponding closed curve and stable point. Since the activities of four populations cannot be shown on one 3D diagram, we show thalamo-cortical bistability on both diagrams (A and B) using different sets of co-ordinates. Bistability in the thalamic (TC-RE) network was obtained by setting the firing of PY and IN cells equal to zero (G S pps), increasing the intrathalamic coupling TC to RE and RE to TC by 35% (c 4 19) and 1% (c 2 c 3 2), respectively and by reducing intra-re inhibition by 5% (Q 6 pps). Bistability in the cortical (PY-IN) loop was obtained by setting

8 474 P. Suffczynski et al. / Neuroscience 126 (24) A 4 Thalamo cortical 3 PY 2 1 B 5 TC 1 Thalamic 1 15 RE Cortical Thalamo cortical PY 2 Fig. 6. Counter-stimulation phenomenon. (A) Output epoch showing paroxysmal oscillations initiated by the triggering stimulus (4 pps amplitude, 1 ms duration) delivered to the cortical input around second 1.5. Paroxysmal activity is prolonged when there is no counterstimulation pulse applied afterward. (B) Output epoch showing paroxysmal oscillations initiated by the triggering stimulus as in A. The triggering stimulus is followed by a counter-stimulation pulse (4 pps amplitude, 1 ms duration) delivered to the cortical input 62 ms after the trigger. In this case only one period of simulated paroxysmal discharge is present and the paroxysm is aborted. The inset displays the output signal and both triggering and counter-stimulation pulses in an extended time scale. It shows that the stimulus effective in annihilating the paroxysmal rhythm was delivered when the PY population was maximally inhibited. (C) Amplitude-phase analysis of the effective counter-stimulus. The plot in polar co-ordinates shows two patches, one (light gray, marked with a sign) corresponding to positive pulses, the other (dark gray, marked with a sign) corresponding to negative pulses. Each patch shows the counter-stimulation parameters (amplitude and phase) that were effective in aborting paroxysmal oscillations. Zero phase corresponds to maximal negative polarity of the output signal. Patch plots were obtained by model simulations with noise components removed and paroxysmal oscillations triggered by an external stimulus as in A, B. The counter-stimulus had 1 ms duration, its phase was sampled from 36 in steps of 1 (assuming cycle frequency 9.28 Hz) while its amplitude was sampled from to 15 pps in steps of 1 pps. The counter-stimulation was delivered after nine cycles of the paroxysm (approximately 1 s) to avoid transient effects influencing the analysis. the output of TC and RE cells equal to zero (G (TC/RE) pps) and by increasing the constant value of cortical input by 25% (P Cx 17.5 pps). Bistability in the thalamo-cortical network depicted in Fig. 7 was obtained using the reference set of parameters. In all simulations the noise component was removed from both inputs for clarity. DISCUSSION Basic properties of the model 1 5 TC 1 15 Fig. 7. Phase diagrams showing three distinct scenarios of bistable behavior in the thalamo-cortical network. For three different parameter settings the coexistence of normal and paroxysmal states in three different neuronal networks (i.e. cortical, thalamic and thalamocortical) is shown. For each parameter setting, two simulations were performed with different initial conditions, one leading to normal, the other leading to paroxysmal behavior and subsequently two attractors were plotted. The closed curve corresponds to paroxysmal activity while the black dot enclosed inside the curve corresponds to the coexisting normal state. (A) The axes correspond to neuronal activities (firing densities in pps) of three neuronal populations (TC, RE and PY). Thalamic attractors are entirely contained in the TC-RE plane while activity at PY co-ordinate is equal to zero. This shows that activity of PY cells does not necessarily contribute to generation of bistability in the thalamic network. (B) The axes correspond to neuronal activities (firing densities in pps) of TC, IN and PY populations. Cortical attractors are entirely contained in the PY-IN plane while activity at TC co-ordinate is equal to zero. This shows that activity of TC cells does not necessarily contribute to generation of bistability in the cortical network. (A and B) Thalamo-cortical paroxysmal oscillation involves interactions between all four neuronal population since paroxysmal attractor involves non-zero activities at all four activity axes (which, however cannot be shown on one 3D diagram). Parameters changes (with respect to the reference set) sufficient to simulate the three distinct bistable networks are given in the text. We constructed a model of interconnected thalamic and cortical populations with parameters directly related to synaptic and cellular properties in order to investigate the mechanisms of generation of synchronized oscillations in a thalamo-cortical neuronal network. We used animal data obtained in rat genetic models of absence epilepsy (WAG/ Rij and GAERS) as the basic experimental data to be modeled. The simulation studies of the dynamical behavior of the model and of its sensitivity to a number of basic parameters led to the following main results: (i). Paroxysmal discharges characterized by 9 Hz large amplitude oscillations represent bifurcations that occur in a neuronal network with bistability properties. This means IN 1 2

9 P. Suffczynski et al. / Neuroscience 126 (24) that two stable states co-exist, one corresponding to the normal on-going EEG activity and the other to the paroxysmal oscillations, and that the system may undergo transitions from one state to another. These transitions can emerge spontaneously and are not induced necessarily by any parameter changes. (ii). The distributions of lengths of paroxysmal and normal on-going epochs are exponential, indicating that transitions between these two stable states occur randomly over time and that the probabilities for the transition between both states can be defined. (iii). Probabilities of transitions between normal ongoing neuronal activity and paroxysmal oscillations depend on a number of model parameters; therefore these probabilities can be controlled in a not unique way. (iv). Paroxysmal oscillations can be annihilated by a well-timed pulse. (v). Since random fluctuations in control parameters and/or dynamic variables can lead to the sudden onset of large amplitude paroxysmal activity, this implies that the occurrence of this type of phenomena is unpredictable per definition. Time distributions of paroxysmal events Here we discuss how our model results compare with experimental findings on rat models of absence epilepsy. A first observation is that paroxysmal activity in rats arises abruptly with a change of signal s amplitude to large values (van Luijtelaar and Coenen, 1986; Snead et al., 1999). This sudden appearance of paroxysmal activity is consistent with the hypothesis that underlying neuronal populations behave as bistable systems. In monostable systems oscillations usually build up since gradual parameter changes are required to reach the paroxysmal oscillatory state. The evidence for bistability in both GAERS and WAG/Rij rats exists, since SW discharges are immediately interrupted by strong sensory (e.g. acoustic) stimulation (Drinkenburg et al., 23;Snead et al., 1999). The role of random fluctuations in triggering the transitions between normal and paroxysmal state in WAG/Rij rats is demonstrated here by the histograms of durations of both ictal and inter-ictal periods of activity (Fig. 4, lower panel). The exponential shape of both distributions indicates that in these animals seizure initiation and termination are random processes with constant probabilities over time. However, we should note that the experimental rat data analyzed here were obtained under vigabatrin, a condition where there is an increase of the occurrence of SW discharges, an increase of their mean duration and a decrease of their peak-frequency as compared with saline conditions (Bouwman et al., 23). These experimental distributions provide proof of existence of the bistability phenomenon predicted in our computer model studies. Another confirmation of our predictions comes from analysis of recordings of GAERS in drug-free conditions (Van Hese, Department of Electronic and Information Systems, Ghent University, Belgium, personal communication). In some rats the distributions of ictal and seizure-free epochs are exponential while in other they can be described by distributions (generalization of exponential distribution) which are also typical for random transitions in bistable systems in more general conditions (Suffczynski et al., unpublished observations). Therefore, we do not exclude that other physiological mechanisms such as progressive and persistent activation of the hyperpolarization-activated depolarizing (I h ) current in TC cells (Bal and McCormick, 1996; Luthi and McCormick, 1998, 1999), or a change in the ionic composition of the extra-cellular space (Pumain and Heinemann, 1985; Amzica et al., 22), may play also a role in the termination or initiation of a paroxysmal burst. In addition to the fact that the present model explains the dynamics of spontaneous appearance of SW seizures, the model study allows also the identification of a neuronal mechanism underlying these dynamical properties. Random fluctuations, may arise due to a number of reasons including quantum noise in opening and closing of ionic membrane channels, axons membrane voltage fluctuations and the probabilistic nature of synaptic transmission (Lopes da Silva et al., 1997). Bistability in our thalamocortical network model results from the specific properties of GABA B receptors and I T current in TC neurons. GABA B receptors, having nonlinear activation properties, require a strong stimulus intensity to evoke a response. Accordingly, in the normal state GABA B receptor-mediated inhibition is not active. Activation of GABA B receptors in TC cells results in prolonged hyperpolarization which is effective in removing the inactivation of the I T current in these cells leading to increased burst firing of TC population. Large TC bursts strongly activate RE cells-directly and indirectly (through PY cells) which results in subsequent activation of GABA B receptors and this cycle tends to repeat itself. Failure to reinitiate the cycle results in seizure cessation. With respect to the essential role of the nonlinear characteristics of GABA B -mediated inhibition, our model is not much different from those of Destexhe (1998, 1999). In the latter model (Destexhe, 1998) the intact thalamic circuit could be forced into either approximately 1 Hz spindle or approximately 3 Hz oscillations depending on the strength of the 3 Hz cortico-thalamic stimulation. This property was provided by the characteristics of GABA B mediated inhibition. Furthermore, increased cortical excitability which caused an increase of discharge in thalamic projecting neurons forced the thalamus to display the 3 Hz paroxysmal discharges which became generalized in the whole thalamo-cortical network. In the latter model, however, bistability was not reported. Another difference with the above models is that in our model increased excitation of PY neurons may activate GABA B receptors in the cortical IN resulting in cyclic interactions in the cortical module at a frequency of SW oscillations (Fig. 7), which may also contribute to the paroxysmal activity in the thalamo-cortical loop. Blocking of GABA B receptors at the cortical level, results in a decrease of amplitude of paroxysmal oscillation in the PY cells (not shown), which suggests that both intrathalamic and intracortical oscillatory mechanisms may contribute to the pathological SW discharges in the model. Interestingly, an increased expression of both GABA B1 and GABA B2 receptor subunits in both the cortex and thalamus

10 476 P. Suffczynski et al. / Neuroscience 126 (24) has been found in GAERS compared with control rats (Princivalle et al., 23). Sensitivity of model performance to parameters: GABA A inhibition, interneuronal synchronization and calcium currents Parameter analysis of the model network revealed that seizure duration is most sensitive to (a) cortical GABA A inhibition (Fig. 5A), (b) to the slope of the sigmoid transfer function of cortical IN (Fig. 5B) and (c) to changes of the Ca 2 currents in RE and TC neurons (Fig. 5C, D). We examine next a number of experimental findings that are in line with these predictions of the model. (a) The sensitivity to the cortical GABA A inhibition is consistent with experimental data obtained in animal models of absence epilepsy. Investigations of the primary abnormalities underlying non-convulsive generalized seizures in GAERS rats revealed, among others, an impairment of GABA A -mediated transmission in the neocortex (Spreafico et al., 1993). Similarly, the cortical hyperexcitability in WAG/Rij rats was demonstrated to be due to a decrease in GABA-mediated inhibition (Luhmann et al., 1995). Injection of GABA A antagonists such as penicillin or bicuculline to the cortex produced SW discharges in cat (Gloor et al., 199; Steriade and Contreras, 1998). Powerful control of cortical excitability by intracortical GABA A inhibition was also demonstrated in vivo and in a modeling study by Contreras et al. (1997). (b) The critical dependence of paroxysmal activity on the slope of the sigmoid transfer function of cortical IN has not a straightforward interpretation. We may state that an increase of the slope of the sigmoid transfer function is directly related to narrowing the distribution of firing thresholds in a neuronal population and thus it represents an increased synchrony in that population. We hypothesize that the slope parameter may mimic the strength of gapjunctional connections within a population of IN. Therefore it may be of interest to note that (i) it was proposed by Velazquez and Carlen (2) that hyperventilation, which reduces blood CO 2 levels and causes alkalosis (Foster et al., 21), may rapidly enhance gap-junctional communication and neural synchrony and (ii) anatomical data indicate that gap junctions in the neocortex are specifically formed among inhibitory cells (Galarreta and Hestrin, 21). In the light of these two observations, our modeling results showing that an increase of the slope parameter has a powerful effect on paroxysmal activity may be related to an increase of electrotonic coupling due to hyperventilation, since the latter is well-known activation method of absence seizures in epileptic patients (Niedermeyer and Lopes da Silva, 1999). (c) A decrease of a calcium current in the RE population (i.e. a decrease of parameter G (RE) ) was found to decrease seizure duration and increase intervals between seizures (Fig. 5C). Our model results are compatible with the experimental findings in epileptic GAERS rats. Tsakiridou et al. (1995) have demonstrated that in these epileptic animals the low-threshold (I T ) calcium conductance in the RE nucleus neurons is elevated in comparison to nonepileptic controls. An increase of T-current is detectable prior to the time of first SW appearance, which may suggest that SW expression in GAERS requires a fully developed network (Tsakiridou et al., 1995). In the same strain of epileptic rats, a pharmacological reduction of burst firing in RE nucleus, attributed to a decrease of the I T calcium current and consequent decrease of a calcium dependent potassium current, resulted in a decrement of paroxysmal discharge duration (Avanzini et al., 1992). Thomas and Grisar (2) put forward an interesting hypothesis that the increased synchrony of the thalamic network, due to an increase of I T current conductance in the RE neurons, may be related rather to a phase-shift in the activity of the TC and RE neurons than to an increase of the amplitude of LTS in RE cells, since the latter was unaffected by I T conductance changes. We also found in our model that an increase of I T current in the RE population changes the phase relation between TC and RE neurons, increasing the network synchrony as indicated by the enhancement of the peak in the power spectrum of the thalamic signals (not shown). Simulation of effects of antiepileptic drugs The model allowed also to investigate the mechanism by which antiepileptic drugs may affect the threshold for seizure occurrence. The most selective anti-absence drug ethosuximide is believed to exert its antiepileptic effect by antagonizing the burst firing in the TC neurons either by decreasing the I T current (Coulter et al., 1989, 199, 1991) or by acting on the noninactivating Na current and on a Ca 2 -activated K current in thalamic cells (Leresche et al., 1998). However, as a more recent study suggests, anti-absence action of ethosuximide cannot be fully explained by the mechanisms at the thalamic level (Richards et al., 23). In our model, a reduction of the amplitudes of LTS generated in TC population (reflected on parameter G (TC) ) results in a decrease of paroxysm s duration and an increase of in-between seizure intervals (Fig. 5D). We should note, in this context, that it is not yet clear to what extent TC cells display LTS during SW oscillations. In WAG/Rij rats Staak and Pape (21) reported that no silent TC cells were observed during SW discharges in vivo, while Steriade and Contreras (1995) found in the cat that about 6% of TC cells were completely silent, and Pinault et al. (1998) and Crunelli and Leresche (22b) pointed out that in GAERS rats the majority of TC cells (9%) do not show LTS during SW discharges. Be as it may, we do not yet know in quantitative terms what may be the contribution of a small percentage of TC cells displaying LTS to the generation of paroxysmal oscillations. The results of Fig. 5D suggest that even a relatively minor change of the contribution of LTS in TC cells may influence significantly the mean paroxysmal epoch duration. We do not exclude, however, other mechanisms of generation of paroxysmal oscillations. Cortical SW has been observed in athalamic cats (Steriade and Contreras, 1998) and studied in computational models (Destexhe et al., 21; Timofeev et al., 22) while thalamic paroxysmal oscillations have been observed in vitro (Bal et al., 1995a,b) and investi-

11 P. Suffczynski et al. / Neuroscience 126 (24) gated in model studies (Destexhe et al., 1996; Lytton et al., 1997). Our model also shows three distinct conditions that are sufficient to generate paroxysmal activity in cortical and thalamic loops separately, and in the complete thalamo-cortical loop (Fig. 7). In the present study we focused, however, on the latter (i.e. thalamo-cortical) possibility since in rat models of absence epilepsy the functional integrity of both cortex and thalamus is required for generation of SW discharges as suggested by lesion studies (Vergnes and Marescaux, 1992; Meeren, 22). The effect of benzodiazepines, such as clonazepam, is related to specific cellular targets mainly within the thalamic networks. Indeed the RE and TC neurons do not have the same kind of GABA A receptors; those of the former have molecular subunits with binding sites for benzodiazepines, in contrast to the latter (Browne et al., 21). Thus these anti-absence drugs are believed to enhance GABA A -mediated inhibition within RE nucleus (Huguenard and Prince, 1994) but not in TC cells, and in this way to attenuate GABA-mediated inhibition of the RE to TC neurons and thus prevent absence seizures. This hypothesis is confirmed by our model since an increase of inhibitory strength between RE cells (reflected on input Q) leads to a decrease of RE output and antagonises paroxysmal activity (Fig. 5E). These modeling results are also in agreement with an experimental study showing that suppression of GABA A inhibition in RE nucleus of GAERS led to an increase in duration of SW activity (Aker et al., 22). Additionally, antiabsence action of clonazepam may include augmentation of GABA A currents in rat cortical neurons (Oh et al., 1995), which, as discussed above, is also consistent with our results (Fig. 5A). Simulations of transition between wakefulness and sleep Our results are also in agreement with observations in the WAG/Rij rats, that the SW seizures predominantly occur during drowsiness and light non-rem sleep (Coenen et al., 1991; Drinkenburg et al., 1991) where sleep spindles are prevalent. In the model we found (Fig. 5F) that a decrease of cholinergic activation from the brainstem, which mainly underlies the transition from the waking state to sleep (McCormick and Bal, 1997), facilitates the generation of paroxysmal activity. The latter result is also in line with the behavioral observation that paroxysmal discharges in GAERS occurred primarily when attention and activity were reduced (Snead et al., 1999) and with the anti-burst action of cholinergic thalamic input as proposed by Buzsáki et al. (199). Does the model account for human absences? The model was developed to account for absence rat data but it is interesting to ask the question whether the model is likely to be relevant to human absence seizures. Human absences and rat absence-like nonconvulsive seizures share many common features like presumed genetic origin, behavioral and EEG patterns and pharmacological profile, but there are also some differences. In typical human absence epilepsy seizures start in childhood and usually disappear in adulthood. In epileptic rats seizures appear in adult animals and continue to occur through the entire life. Another difference is that the SW frequency among humans is between Hz while among absence epilepsy rats the frequency varies from 7 11 Hz. This difference in frequency can be easily accounted for by minor parameter changes in the model: e.g. a reduction of GABA A conductance and an increase of GABA B conductance in PY and TC cells and a small increase of membrane time constant in all types of cells can transform the model s output into paroxysmal episodes of frequency 4 Hz arising also from small amplitude, waxing and waning background of dominant frequency about 9 1 Hz. These results are similar to those of Destexhe (1999) showing that thalamo-cortical circuit can generate both fast (5 1 Hz) and slow (approximately 3 Hz) SW oscillations depending on balance between GABAergic conductances in thalamic relay cells. Since the long term properties of our model apply also for the parameter setting with the characteristics of human SW frequencies, we may predict that the distributions of absence seizure durations and inter-ictal intervals in humans should be exponential as they are in rats. Interestingly, exponential shape of histograms of durations of ictal events has been found in human recordings from absence patients (Suffczynski et al., unpublished observations) and in other types of human epilepsy (Velazquez et al., 1999). Can absence seizures be predicted? Following the hypothesis that absence-like seizures in rats and human absences correspond to the same basic mechanisms, an interesting question arises whether the model can help to understand whether the occurrence of absence seizures may be predicted some time in advance or not. The fact that we found that random fluctuations can be responsible for the sudden onset of the paroxysmal oscillations implies that the occurrence of these seizures cannot be predicted since fluctuations are unpredictable by definition. This conclusion is consistent with classic clinical observations in humans as expressed by Lennox and Lennox (196): If warning occurs, the diagnosis of petit mal may be questioned. While the possibility of prediction of the exact timing of absence seizures is rejected by our model, quantifying the probability for an instantaneous seizure transition, e.g. by indirectly measuring some control parameters, might still be a realistic option. Can paroxysmal oscillations be aborted? In a bistable systems where a stable steady state coexists with a stable limit cycle the transitions between the two can be triggered by a single pulse. In addition, the oscillation may be terminated if a brief stimulus of sufficient magnitude is delivered at a critical phase of the oscillation (Glass and Mackey, 198). The results of the present model confirm this theoretical prediction as shown in Fig. 6A, B.A detailed analysis of the effective pulse parameters is shown in Fig. 6C. It shows that paroxysms can be aborted either by an excitatory pulse delivered before the time at which the main PY cells population reaches maximal neg-

12 478 P. Suffczynski et al. / Neuroscience 126 (24) ative polarity or by an inhibitory pulse delivered before the time at which the main population is maximally depolarized. Experimental application of these results may be feasible, although in biological networks spatially distributed degrees of freedom must be taken additionally into account. Model s scope and limitations Modeling is unavoidably based on simplifications of real systems. In our view, the simpler the model that can explain a particular phenomenon, the more fundamental is its scope. We consider model building an iterative process. We started with the basic features of the system to be modeled and we added subsequent known mechanisms such that the model was sufficient to account for a set of specific experimental results. Nevertheless, a simple model has obvious limitations resulting from simplifications of the real system. In the present work we followed the lumped-circuit approach where all neurons of a given type are lumped together into a representative element, a population, and the distributed neuronal network is compressed into a small network of interconnected lumps. This approach does not allow to investigate interactions between single cells of given types. In addition, since we modeled single cortical and single thalamic modules we could not investigate spatial dynamics. An important issue concerns the nature of a model lump. Our intuitive definition for a neuronal lump is a group of cells of a given type that are spatially close to each other. Realistic modeling of neural systems has to incorporate stochastic components. In the present model Gaussian noise is introduced into the system from outside, i.e. by means of fluctuating inputs. In addition, one could consider noise sources arising in internal variables like synaptic conductances, resting potentials or firing densities. Our approach is motivated by computational simplicity and does not limit the generality of the results. Since external noise propagates to all dynamic variables of the system it mimics internal noise sources. However, our approach can limit the validity of results related to dynamical correlation between model variables. Our model assumptions also limited the range of frequencies available in the model s output. The lower limit is bounded by the slowest process included in the model. It appears to be the duration of GABA B postsynaptic potentials which last about 5 ms, giving lower frequency bound about 2 Hz. The upper limit comes from the simulation of average firing rate instead of single action potentials. Since the averaging domain is approximately 2 ms, as commonly assumed (Arbib, 1998), it imposes a theoretical upper limit, of about 5 Hz. We followed a lumped modeling approach, but we considered necessary to include some processes at the level of neurons membrane dynamics that are responsible for burst generation in thalamic cells. However we modeled average burst generation in a simplified manner by taking into account basic properties of I T calcium current, but not other currents like hyperpolarizationactivated (I h ) current or calcium dependent potassium currents (I K(Ca) ) that also contribute to burst generation. These simplifications in TC cells were based on modeling work of Wang et al. (1991) who showed that T-type Ca 2 current together with leakage current are sufficient to describe the LTS. On the other hand the inclusion of I h current, and its dependence on intracellular Ca 2, could change the model s on going behavior from continuous to periodically recurring spindle activity (Destexhe et al., 1993; Zygierewicz et al., 21). I leak and I T currents were sufficient to simulate LTS generation in RE cells in the context of the present study. Addition of I K(Ca) current would be necessary to simulate intrinsic oscillations in those cells (Bal and McCormick, 1993). An inclusion of additional voltage- and Ca 2 -dependent currents could, of course, influence the occurrence of paroxysmal activity and corresponding duration distributions, as discussed in the first section of the Discussion. In the present model we also did not include N-methyl- D-aspartate (NMDA) type of receptors since preliminary simulations with NMDA receptors included in the cortical module showed no qualitative difference in model s behavior. Similar conclusion was reached by Destexhe (1998). Finally, we made a number of simplifying assumptions regarding model parameters. E.g., the time constants and amplitudes of postsynaptic currents of a given type and amplitudes of leak currents and membrane capacitances were assumed to be equal in both thalamic and cortical populations. Also the characteristics of the sigmoid functions in the PY and IN populations were assumed to be the same. While the simplifications described above do not appear to have pronounced consequences for model s main features, they significantly limit the number of model s parameters, thus simplifying model analysis. We can argue that our model, inspired by neurophysiological observations while constructed with low computational complexity, has allowed us to test a number of hypotheses and the formulation of new ones concerning mechanisms responsible for the spontaneous occurrence of paroxysmal episodes in rat models of absence seizures and their main dynamical properties. Acknowledgements Piotr Suffczynski was partly sponsored by grant 8 T11E 3 17 from Polish Committee for Scientific Research and by Lopes da Silva Fellowship from the Christelijke Vereniging voor de Verpleging van Lijders aan Epilepsie (Christian Society for the Management of Patients with Epilepsy), Heemstede. We thank anonymous reviewers whose comments were valuable for this paper. REFERENCES Aker RG, Ozkara C, Dervent A, Onat FY (22) Enhancement of spike and wave discharges by microinjection of bicuculline into the reticular nucleus of rats with absence epilepsy. Neurosci Lett 322: Amzica F, Massimini M, Manfridi A (22) Spatial buffering during slow and paroxysmal sleep oscillations in cortical networks of glial cells in vivo. J Neurosci 22: Arbib M (1998) The handbook of brain theory and neural networks: a Bradford book. Cambridge, MA: The MIT Press. Avanzini G, de Curtis M, Marescaux C, Panzica F, Spreafico R, Vergnes M (1992) Role of the thalamic reticular nucleus in the

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15 P. Suffczynski et al. / Neuroscience 126 (24) Model description APPENDIX The diagram of the Simulink model is shown in Fig. 8A. It consists of four (PY, IN, TC and RE) interconnected neuronal populations. In each population, incoming impulses generate postsynaptic currents which are integrated giving rise to the mean membrane potential of the population. The latter is transformed to population output (i.e. firing rate) through a nonlinear transfer function. Each element of the model is described successively. Parameter values are summarized in Table 1. The setting of most of parameters values was based on available physiological data. Some modifications of parameter values (e.g. G (TC), G (RE) ) were necessary due to the simplifications that we resorted to in our model. The Simulink (version 5.) is available to any interested reader by request to the author. Synaptic transmission To simulate the time evolution of the transmembrane potential we used the general equation: dv i C m dt i I syn g leak V i i V leak,i TC, RE, PY, IN i I syn where the synaptic currents are: i g syn (1) V i i V syn (2) In (1, 2) V is the membrane potential, C m is the membrane capacitance, g leak is the leak current conductance, g syn is the synaptic current conductance, V leak and V syn are the reversal potentials of leak current and synaptic current, respectively. Synaptic conductances were modeled by convolving incoming action potential sequence, i.e. firing density (pulse syn (t)) with a synaptic impulse response function (h syn (t)): i g syn t t i h syn t pulse syn d (3) h syn t A syn exp( a 1syn t exp( a 2syn t)],a 2syn a 1syn,syn AMPA,GABA A, GABA B (4) The equations (3, 4) can be obtained as solution of the 2nd order ordinary differential equation: d 2 g t dt 2 a 1 a 2 dg t dt a 1 a 2 g t A a 2 a 1 pulse t (5) Following the observation (Kim et al., 1997) that high stimulus intensity is needed to activate GABA B receptor, we assumed that the amplitude of the GABA B postsynaptic current increases nonlinearly with the firing density of the RE and IN populations. The nonlinear activation function of GABA B receptors in RE and IN cells has the form: B F j 1 exp F j B / B 1, j RE, IN (6) Table 2 gives synaptic currents in four neuronal populations (to use with the equations (1,2)) and pulse densities contributing to each synaptic current (to use with the equation (3)): Mean membrane potential to firing rate conversion Conversions from mean membrane potential to pulse densities in cortical populations are of the sigmoidal form (Wilson and Cowan, 1972; Lopes da Silva et al., 1974; Freeman, 1975; and Wendling et al., 22): F k V k G s / 1 exp V k S / s, k PY, IN (7) Conversions from mean membrane potential to pulse densities in thalamic populations took into account the burst firing of thalamic cells that occurs at the hyperpolarized membrane potential levels of these cells. A burst generated by thalamic neuron consists of a LTS mediated by I T calcium current and fast sodium spikes on the top of the LTS. In a population, the relative number of cells that fire LTS at the time t is equal to the proportion of cells in which I T current is de-inactivated and which, at the same time, are depolarized above the threshold for LTS generation. To account for this behavior we introduced new variables n inf (V) and m inf (V). These functions are the steady-state functions of the mean membrane voltage of the population V and express the fractions of cells in which I T current is de-inactivated and activated, respectively, as a function of clamped value of V. If we assume that activation of the I T current is immediate at the appropriate value of membrane potential V, the fraction of cells in which the I T current is activated at time t is given by m inf (V(t)). We also assume that reaching a steady state of I T current de-inactivation is delayed with respect to membrane potential change and this delay is accounted for by convolving the steady state de-inactivation function n inf (V) with a delay function (h n ). Finally, we assume that each LTS triggers a burst of fast action potentials at the frequency G. The pulse density associated with burst firing of the thalamic population is: F (l) V (l) G (l) l m inf V (l) n (l) V (l),l TC, RE (8) n (l) V (l) t h n t n (l) inf V (k) d (9) h n t N exp( n 1 t exp( n 2 t)],n 2 n 1,N n 1 n 2 / n 2 n 1 (1) In analogy with equation (5) equations (9,1) can be represented in a differential form that is more convenient for certain numerical implementations. The activation (m inf ) and inactivation (n inf ) functions of the I T current are of the sigmoidal form: f (l) inf V (l) 1 exp V (l) (l) f / (l) f 1,f m,n,l RE,IN (11) The scheme of the transformation between the mean membrane potential and the average firing rate in a burst mode is depicted in Fig. 8B.

16 482 P. Suffczynski et al. / Neuroscience 126 (24) Fig. 8. Model diagram. (A) Thalamo-cortical network model consisting of interconnected cortical and thalamic modules. The cortical module consists of two interconnected populations of the PY and IN neurons. In both populations membrane leakage, postsynaptic AMPA and GABA A and GABA B (both in the PY population only) currents are assumed to contribute to the mean membrane potential. Changes of mean membrane potential are transformed into firing densities. In both cortical populations the transformations between mean membrane potential and firing density are described by sigmoidal functions. The PY population receives external excitatory input. The model output is the mean membrane potential of the PY population. The thalamic module consists of two interconnected populations of the TC and RE neurons. In the TC and RE populations, membrane leakage and postsynaptic AMPA, GABA A and GABA B (slow GABA B is present in the TC population only) currents contribute to mean membrane potential. The latter is transformed into firing density using transformation functions shown in part B. The TC population receives sensory input while the RE population receives external inhibitory input. Coupling constants c 1 c 13 represent the average numbers of connections between different cell types. (B) Details of the blocks that represent the generation of TC/RE bursts. The latter relate mean membrane potential of the population to the firing density in a burst firing mode. The functions I T activation and I T inactivation describe the fractions of cells in which the I T current is activated and inactivated, respectively, as a function of mean membrane voltage. The impulse response function I T inactivation delay represents the time delay in the fraction of cells in which I T is inactivated following changes in mean membrane potential. The gain G TC/RE represents the frequency of action potentials during a single burst.