External excitatory stimuli can terminate bursting in neural network models

Manuscript submitted to Epilepsy Research, January 2002 External excitatory stimuli can terminate bursting in neural network models Piotr J. Franaszc...
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Manuscript submitted to Epilepsy Research, January 2002

External excitatory stimuli can terminate bursting in neural network models Piotr J. Franaszczuk∗ , Pawel Kudela, Gregory K. Bergey Department of Neurology, Johns Hopkins Epilepsy Center, Johns Hopkins University School of Medicine, 600 North Wolfe Street, Meyer 2-147, Baltimore, MD 21287

Abstract The concept of modulating or terminating seizure activity by brain stimulation is attracting considerable attention. The ability of such external excitatory stimuli to terminate repetitive bursting may depend upon identifiable parameters. We investigate the ability of external stimuli to terminate bursting under various conditions in defined neural network models. Networks of multiple neurons (n = 90), with both inhibitory and excitatory synaptic connections were modeled using conductance-based models with a reduced number of variables. Each neuron in the network has synaptic connections from randomly chosen excitatory and inhibitory neurons. The type and number of connections were kept constant. The initial parameters of the networks were chosen to simulate synchronized repetitive bursting activity. Two basic models of repetitive bursting activity were developed. The first model is a single network with constant random excitatory input, the second incorporates two networks with no random background input, but with a feedback loop with a delay of 600–900 ms connecting two networks. The ability of external excitatory stimuli to terminate repetitive bursting in each model was studied. External excitatory stimulation terminates repetitive bursting in both models; however, only in the models with a feedback loop, is the burst termination long-lasting. In the single network model with constant random excitatory input, termination is of short duration and recurrent bursting resumes. Timing of the application of the stimulus to the dual network model is critical; long-lasting termination occurs only when the stimuli to the first network are during a time when the connected network is still relatively refractory. The delay in the loop determines the exact timing of the stimulus. Synaptic inhibition is not required for burst termination. Neural network models provide systems for the study of burst termination and can help define the requirements for such stimuli to successfully terminate bursts. Studies in these models can guide the development and application of such stimuli in intact systems. Keywords: Neural network; Seizure; Epilepsy; Bursting; Stimulation; Seizure termination

1. Introduction

secondarily generalized epileptic seizures are typically of finite duration; most complex partial seizures last only minutes (Theodore et al., 1983). There are various potential mechanisms for spontaneous seizure termination. Although it may be attractive to postulate a role for GABAergic network inhibition in seizure termination, the contributions of these mecha-

Epileptic seizures represent temporary, episodic periods of increased network excitation with variable propagation. Even in the untreated patient, partial or ∗ Corresponding

author. Tel.: +1-410-614-8770; fax: +1-410-9550751. E-mail address: [email protected] (P.J. Franaszczuk)

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nisms is not fully understood. Indeed GABAA mediated inhibition may be reduced by prolonged or repetitive stimulation (Thompson and Gahwiler, 1989a; Jensen et al., 1999). While GABAB mediated inhibition does not desensitize and can produce slow IPSPs, it may actually contribute to disinhibition by presynaptic inhibition of inhibitory terminals (Thompson and Gahwiler, 1989b; Burgard and Sarvey, 1991). The complete role of GABAB mediated inhibition remains to be elucidated. Intrinsic voltage dependent membrane conductances (e.g. calcium mediated potassium currents) may make more important contributions to burst termination (Lancaster et al., 1991) than GABAergic inhibition. In vitro studies in hippocampal preparations have demonstrated the ability of applied currents to modulate spontaneous epileptiform activity. This has been seen in instances of reduced calcium where synaptic transmission is abolished (Warren and Durand, 1998; Ghai et al., 2000). High frequency sinusoidal electric fields can temporarily suppress epileptiform activity in hippocampal slices in both low Ca2+ and high K+ environments (Bikson et al., 2001). Continuous adaptive electric fields have been used to reduce seizure activity in hippocampal slices (Gluckman et al., 2001). Recently (Bergey and Franaszczuk, 2001), it has been shown that mesial temporal onset seizures in humans are characterized by increasing signal complexity prior to termination. These observations suggest that the dynamic nonlinear characteristics of the epileptic focus may underlie seizure evolution, duration, and termination. Understanding these changes in seizure dynamics can help explain seizure termination without requiring (or excluding) specific synaptic or membrane mechanisms. There is growing interest in chronic brain stimulation to reduce seizures. Vagus nerve stimulation (VNS) (Ben-Menachem, 1996; Labar et al., 1999; Morris and Mueller, 1999) has been approved for the treatment of epilepsy and anterior thalamic stimulation (Velasco et al., 1995, 2000) is undergoing human investigation. Each of these modalities employs chronic stimulus parameters to reduce seizure frequency. With implanted VNS, the option exists for manual activation of the stimulus paradigm with a hand-held magnet. This has been reported to abort seizures in some patients, but has not been quantita-

tively studied. Such on demand VNS activation may be very different from direct stimulation of neural networks since it involves activation of remote (e.g. brainstem) polysynaptic inputs. Afterdischarges (AD) produced by cortical stimulation during brain mapping prior to seizure surgery can be terminated by electrical stimulation in some patients if applied early (Lesser et al., 1999; Karceski et al., 2000). These AD represent repetitive or periodic excitatory discharges triggered by external currents. The fact that external electrical stimulation can at times terminate these discharges at least raises the possibility of new methods of seizure control. Already algorithms exist for computerized detection of seizure onset (Gotman, 1999; Lesser and Webber, 2000). Indeed investigators have used such detection paradigms to trigger automated drug delivery to reduce focal epileptiform activity in experimental models (Stein et al., 2000). Other detection algorithms (Echauz et al., 2001; Esteller et al., 2002) are being implemented in responsive systems to terminate spontaneous seizures in humans (Bergey et al., 2002). We report here two neural network models that can reproduce characteristics of repetitive synchronous bursting. Patterns of seizure activity in humans are dynamic. A single seizure may sequentially include periodic spike activity (single spikes repeated in regular intervals), organized rhythmic activity (the large amplitude predominantly monotonic rhythmic activity monotonously changing frequency) and intermittent bursting (periods of polyspike activity alternating with periods of diminished activity) (Franaszczuk et al., 1998). We have chosen the pattern of intermittent bursting as a framework for the models here. This pattern is often seen prior to spontaneous cessation of seizures in humans; similar patterns can be seen in humans after discharges produced by cortical stimulation. Using these models we have investigated the ability of external stimulation to terminate bursting activity, including experimental paradigms where there is no network inhibition. Preliminary results were presented in abstract form (Franaszczuk et al., 1999, 2000). 2. Methods The models of synchronous bursting in a reduced neural network are based on the principles utilized in previously described models (Kudela et al., 1997, 2

1999, 2002) similar to the models developed by Traub and Wong (Traub and Wong, 1981, 1982). In our previously reported models only excitatory connections were modeled, whereas in the present model, inhibitory synaptic connections can also be incorporated. In most simulations 10 % of the neurons are inhibitory (Traub and Miles, 1991), although some studies were done in networks with exclusively excitatory connections (by setting weights for inhibitory connections to zero). The principles of these networks incorporate active membrane properties for each neuron, defined numbers of synaptic connections for each neuron, and modifiable synaptic weights for excitatory and inhibitory connections respectively. Each neuron is modeled as a single compartment. This results in considerable benefits in computational efficiency when dealing with networks of more than a few neurons. Neurons are modeled as single compartment units using modified Av-Ron–Rinzel’s (Rinzel, 1985 AvRon et al., 1993, 1994) reduced model equations. These modeled neurons are synaptically connected (see below) and capable of generating action potentials. The neuron model incorporates two inward currents, INa and ICa , three outward potassium currents, the delayed rectifier IK , the Ca-dependent IK(Ca) and the transient IA current, and a leak current IL . The synaptic connection between cells is modeled by a synaptic current Isyn . The synaptic conductance is represented by a sum of two exponential functions. The overall strength of a connection is represented by a single synaptic weight parameter and a delay parameter represents all delays between cells. In these simulations we use a network of 81 excitatory cells and 9 inhibitory cells to simulate a small locally connected region of the brain tissue (Fig. 1). Each cell receives excitatory input from two cells and inhibitory input from three cells. Presynaptic neurons are chosen randomly from all 81 excitatory and 9 inhibitory neurons respectively. A pseudo-random generator is used to choose connections for each cell. This produces a network with no predefined structure of, with a loose recurrent excitatory and inhibitory connections. The weight of synaptic connections is equal to 60 for intrinsic excitatory connections and -120 for inhibitory connections for the most of the simulations. These values were chosen after multiple simulations with different relative synaptic weights, so that synchronized network bursting could be produced by excita-

I

E

Fig. 1. Schematic diagram of a randomly connected local network of 90 neurons. The network consists of 81 excitatory neurons (open circles) and 9 inhibitory neurons (filled circles). Each neuron has two excitatory inputs and three inhibitory inputs. All output connections for one inhibitory (I) and one excitatory neuron (E) are shown.

tory input. In some simulations the weights were normally distributed with S.D. 6 and mean 60 or -120 for excitatory and inhibitory synapses respectively. The interneuronal latency is 3.6 ms for most of the simulations. Some simulations were done with interneuronal latencies normally distributed with the mean equal to 3.6 ms and S.D. equal to 0.05 ms. Two distinct models of network activation were implemented. In the first model the network was activated by continuously applying random (Poisson) action potentials to excitatory synapses (weight = 120) of four selected neurons. Applying random action potentials to just one neuron if this neuron and its postsynaptic neurons have sufficient connectivity can activate this type of network. Activation of four randomly chosen neurons in a 90 neuron network assures that in the majority of simulations at least one of these neurons will initiate bursting in the network. After bursting had been established the external depolarizing current or large random background input was applied to all neurons in the network. In the dual networks model (Fig. 2) four randomly chosen neurons from one network (A) are synaptically connected to four selected neurons from the second network (B). Similar to the single network model, choosing four neurons assures that almost all neurons in the network B will be activated. An additional loop from one neuron in network B to one neuron in network A, with a 600–900 ms delay, is included to sim3

ulate a traveling wave in a large network. In this case the neurons were not chosen randomly. The neuron in network B was chosen to avoid the ’quiet neuron’ not bursting with other neurons. More than 80% of excitatory neurons in this randomly connected network could be chosen for this purpose. The output neuron of the loop in the network A is the one with proper connections to be able to activate the network A. More than 70% of excitatory neurons fulfill this requirement. In this case the network was activated only once at the start of the simulation, by applying an external current (Iext = 10 µA/cm2 ) to all neurons in network A. After bursting had been established the external depolarizing current of the same amplitude was applied to all neurons in network A. Thus, the first model is simulating an isolated locally connected network, while this second model represents locally connected networks interconnected in a larger structure.

2.1. Neuron model equations

dV = Iext − INa − ICa dt − IK − IK(Ca) − IA − IL − Isyn

Cm

dW W∞ (V ) −W = dt τW (V ) dX X∞ (V ) − X = dt τX dC = K p (−ICa ) − RC dt dB B∞ (V ) − B = dt τB where τW (V ) = W )] W )] −1 1  [aW (V −V1/2 [−aW (V −V1/2 e +e γ  P )] −1 [−2aP (V −V1/2 P∞ (V ) = 1 + e

external stimulus A

for P ≡ W, m, X, A, B

B

V is the membrane potential, W is the recovery variable, C is the intracellular calcium concentration, X and B are respectively the calcium channel activation variable and the transient potassium channel inactivation variable. The steady-state functions m∞ , A∞ , W∞ , X∞ , and B∞ are modeled as sigmoidal curves, determined by two parameters: the half maximum voltage V1/2 (values are -31, -20, -35, -45 and -70 mV respectively) and a slope aP of the curve at this point (values are 0.065, 0.02, 0.055, 2.0, and -0.095 respectively). Kp = 0.0002 is the conversion factor from calcium current to concentration and R = 0.006 is the removal rate constant of the intracellular calcium concentration. Cm = 1 µF/cm2 is the membrane capacitance. Here τW (V ) is the relaxation time function for the recovery variable W ; τX = 25 ms and τB = 10 ms are the relaxation time constants for calcium activation variable X, and potassium transient inactivation variable B respectively. Ion currents Ii are described by the product of three terms: the maximal conductance gi , the activation and inactivation variable or function, and the driving force (V −Vi ).

delayed feedback loop

Fig. 2. Schematic diagram of the dual network model. Each square represent a locally connected network of a type illustrated in Fig. 1. Four neurons in network A are connected by excitatory connection with four neurons in network B. Network B is connected with network A by a single delayed excitatory connection. An external stimulus is applied to all neurons in network A to activate or terminate the bursting activity.

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of decreasing the error associated with determining the time of occurrence of action potentials.

INa = gNa m3∞ (V )(1 −W )(V −VNa ) KC ICa = gCa X 2 (V −VCa ) KC +C IK = gKW 4 (V −Vk ) C IK(Ca) = gK(Ca) (V −Vk ) Kd +C IA = gA A∞ (V )B(V −VK ) IL = gL (V −VL )

3. Results The parameters of neuron and synaptic models were selected to be consistent with observations from in vitro and in vivo systems and to allow generation of bursts of action potentials in a small network. In the single network model, the network was activated by applying random excitatory input to four selected neurons. Outputs from all neurons in the network in a typical simulation are shown in Fig. 3A; the input produces a clear pattern of recurrent relatively synchronous bursting activity. Fig. 3B shows the membrane potentials from three neurons enlarged to reveal detail with both action potentials and inhibitory and excitatory postsynaptic potentials. Although the synaptic connections were chosen randomly, the presence of repetitive bursting activity does not depend upon a specific set of connections and was observed in all instances. Application of a depolarizing current at a certain level (Iext = 10 µA/cm2 ) to all neurons temporarily alters bursting activity (Fig. 4). The stimulus, applied at the onset of a synchronized burst, results in transient increased synchrony, but fails to terminate recurrent bursting, producing only a temporary delay manifest by increased interburst interval. Even increasing the stimulus 20-fold (to Iext = 200 µA/cm2 ) again only temporarily alters bursting activity. Indeed above a the current of (Iext =10 µA/cm2 ), which is sufficient to trigger action potentials, the amplitude of the stimuli did not change the response of the network. Application of the stimulus at other times (e.g. after a burst) also only results in a temporary disruption of repetitive bursting. In the model with a long delay excitatory feedback loop, behavior of the network differs significantly. In this model, two 90-neuron networks are connected through four excitatory connections between selected neurons (Fig. 2). Instead of receiving random excitatory input, one selected neuron in network A receives excitatory input from an feedback loop from network B with a delay of 800 ms. Initially the network is quiet. After application of a relatively weak stimulus (Iext = 10 µA/cm2 ) to network A, both networks generate bursts at regular intervals determined by the delay in the feedback loop (Figs. 5 and 7). The termination of each burst in any cell is regulated by the calcium

where gNa = 120 mS/cm2 , gCa = 1.0 mS/cm2 , gK = 15 mS/cm2 , gA = 12.5 mS/cm2 , gL = 0.3 mS/cm2 , gK(Ca) = 3.5 mS/cm2 are maximum conductances for the respective channels and VNa = 55 mV, VCa = 124 mV, VK = -72 mV and VL = -50 mV are values of the reversal potentials for the respective ions and leak current. Kd = 0.5 and KC = 2 are the calcium concentration functions constants. 2.2. Synaptic model equations Nsyn

Isyn (t) =

∑ w j g j (t − τ j )(V − Esyn )

j=1

N

g j (t) = g0 ∑ (e−(t−ti )/τd − e−(t−ti )/τ0 ) i=1

where index i denotes past action potentials and j input synapses. τ j = 3.6 ms is the total delay for synaptic connection j, g0 = 0.0066 mS/cm2 , is the synaptic conductance constant, g j (t) is the conductance function and Esyn is the synaptic reversal potential equal to -10 mV and -72 mV respectively for excitatory and inhibitory synapses. τd = 3 ms and τo = 0.5 ms represent respectively decay time and onset time constants of a PSP. Synaptic weight w j is modeled as an integer 60 for local excitatory connections and -120 for local inhibitory connections. Excitatory connections between networks and random external excitatory connections have weight w = 120. ti denotes time of arrival of the i-th action potential at the synapse, N is the number of past action potentials with significant contribution to the sum and Nsyn is the number of synaptic inputs. The ordinary differential equations were solved numerically using the forward Euler method with a time step of 0.01 ms. We determined that with this time step the forward Euler method is stable for our equations. Choosing a small step gives additional benefit 5

A

→ →





1s

B

25 mV

400 ms

Fig. 3. Illustration of the activity in the network illustrated in Fig.1. A. Traces of simulated membrane potentials of all 90 neurons are shown. The top 81 traces are for the excitatory neurons and the bottom 9 (marked by vertical dotted line) are for the inhibitory neurons. Four selected neurons (marked by arrows at left) are continuously activated by random Poisson excitatory synaptic input, with weight w = 120 and Poisson process λ = 0.00003. B. The membrane potentials from three neurons enlarged from the box in panel A reveal detail with both action potentials and inhibitory and excitatory postsynaptic potentials.

6

→ →





1s

External stimulus

Fig. 4. Illustration of the activity of the same network as in Fig. 3 with application for 50 ms of an external depolarizing current Iext = 10 µA/cm2 to all neurons. The stimulus (arrow) triggers a synchronized burst, alters the pattern of activity, but does not terminate activity permanently. The top 81 traces are for the excitatory neurons and the bottom 9 (marked by vertical dotted line) are for the inhibitory neurons. Four selected neurons (marked by arrows at left) are continuously activated by random Poisson excitatory synaptic input, with weight w = 120 and Poisson process λ = 0.00003. The one second horizontal time bar is for calibration.

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Network A

External stimulus

1s

Network B

Fig. 5. Illustration of the results of the simulations of the two networks with an external feedback loop. The four selected neurons from network B have inputs from the four selected neurons in the network A. A feedback loop is modeled as the excitatory connection from one neuron in network B to one neuron in network A, with a delay of 800 ms. The bursting activity is initiated by applying a 50 ms external current (Iext = 10 µA/cm2 , arrow) to all neurons in the network. After the external stimulus, both networks are generating bursts periodically with the interburst interval determined by the delay in the feedback loop. The top 81 traces for each network are for the excitatory neurons and the bottom 9 (marked by vertical dotted line) are for the inhibitory neurons. The one second horizontal time bar is for calibration.

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Network A

1s

External stimulus

Network B

Fig. 6. Illustration of the continuation of the activity from Fig. 5. When the same external current (Iext = 10 µA/cm2 , t = 50 ms) is applied to all neurons in network A at a specified time before excitability has returned to baseline in network A (see text), the bursting of the network ceases. The four selected neurons from network B have inputs from the four selected neurons in the network A. A feedback loop is modeled as the excitatory connection from one neuron in network B to one neuron in network A, with a delay of 800 ms. The top 81 traces for each network are for the excitatory neurons and the bottom 9 (marked by vertical dotted line) are for the inhibitory neurons. The one second horizontal time bar is for calibration.

9

Network A

1s

Network B

Fig. 7. Illustration of the activity of the same epoch after activation as in Fig. 6, but without application of the second stimulus. The periodic bursting activity is occurring continuously. The four selected neurons from network B have inputs from the four selected neurons in the network A. A feedback loop is modeled as the excitatory connection from one neuron in network B to one neuron in network A, with a delay of 800 ms. The top 81 traces for each network are for the excitatory neurons and the bottom 9 (marked by vertical dotted line) are for the inhibitory neurons. The one second horizontal time bar is for calibration.

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Network A

1s

External stimulus

Network B

Fig. 8. Illustration of the activity of the same epoch shown after activation as in Fig. 6, but with application of the second stimulus at a different time. The periodic bursting activity is only briefly altered. The four selected neurons from network B have inputs from the four selected neurons in the network A. A feedback loop is modeled as the excitatory connection from one neuron in network B to one neuron in network A, with a delay of 800 ms. The top 81 traces for each network are for the excitatory neurons and the bottom 9 (marked by vertical dotted line) are for the inhibitory neurons. The one second horizontal time bar is for calibration.

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Network A

1s

External stimulus

Network B

Fig. 9. Illustration of the termination of bursting activity by an excitatory external current (Iext = 10 µA/cm2 , t = 50 ms) similar to Fig. 6, but in purely excitatory networks. The networks are the same as in Figs. 4–7 but all inhibitory connections are deactivated by setting synaptic weight w = 0. The four selected neurons from network B have inputs from the four selected neurons in the network A. A feedback loop is modeled as the excitatory connection from one neuron in network B to one neuron in network A, with a delay of 800 ms. The top 81 traces for each network are for the excitatory neurons and the bottom 9 (marked by vertical dotted line) are for the inhibitory neurons (inactive). The one second horizontal time bar is for calibration.

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regulated potassium current IK(Ca) , which is responsible for cell adaptation to repetitive input excitation. After termination of a burst there is a period when excitability of the cell is diminished. Application of a depolarizing current to all neurons in network A, just before neuronal excitability returns to baseline (i.e. ability to fire repetitive bursts), produces a wellsynchronized burst in this network. However, in network B bursting is not fully developed because of still decreased excitability. In particular, if the neuron in the network B providing output to the loop does not burst, this results in cessation of subsequent synchronized bursts (Fig. 6). Which of the neurons in network B do not burst under these circumstances depends upon the random excitatory connection between the two networks. With various simulations about 40-60 % of the neurons in the network will not burst. Interestingly when there is no functional inhibition the percentage of neurons in network B that do not burst is even higher (Fig. 8). The network does not burst until another trigger for synchronous bursting occurs. This is distinct from the models without the feedback loop incorporated (Fig. 4) where bursting recurs soon after the applied current as a result of random input of action potentials. To illustrate that indeed the application of the stimulus is the cause of the cessation of bursting, we show simulation of the same exact network in the same period without the stimulus in Fig. 7. This simulation was continued for 50 s without any observed change in repetitive bursting activity. However, if the loop delay is reduced to 600–790 ms, a spontaneous cessation of bursting is observed in some simulations. The application of the external current in models with incorporated feedback loops needs to be carefully timed so that the stimulation in the network A coincides with the relative refractory period in the network B. Termination of bursting can be accomplished with the same stimulus intensity used for excitation. In both Figs. 5 and 6 the external stimulus current is the same (Iext = 10 µA/cm2 ). Stimuli that are applied outside of very narrow (∼15 ms duration) window fail to terminate bursting permanently (Fig. 8). The introduction of a moderate (S.D. 6) variability of synaptic weights among neurons does not change these results. Introduction of variability in interneurons delays reduces excitability of the network. With a small variability in delays (standard deviation 0.05 ms) the network can produce recurrent bursting which is easier to termi-

nate. The window for the terminating stimulus is ten times larger then for homogenous networks (∼150 ms duration). The application of the external current in models with incorporated feedback loops needs to be carefully timed so that the stimulation in network A coincides with the relative refractory period in network B. Termination of bursting can be accomplished with the same stimulus intensity used for excitation. In both Figs. 5 and 6 the external stimulus current is the same Iext = 10 µA/cm2 . Interestingly this long-lasting cessation of recurrent synchronized bursting in the loop model is not dependent on the presence of inhibitory connections in the model network. In purely excitatory networks, external currents of a similar intensity can produce long lasting burst cessation as well (Fig. 9), even though the individual bursts are more synchronized between neurons than in the mixed (inhibitory and excitatory) networks. 4. Discussion The models studied here are designed to represent repetitive bursting activity typically seen in the later phases of seizures occurring in the human brain. While the design of the model incorporates specific numbers and weights of synaptic connections, the patterns of bursting produced are reproducible in all such models, occurring in all models with random connections. The two distinct models for repetitive bursting allow for study of differential effects of stimuli. If a seizure results from changes in background activity then the first model may be more representative. If a seizure is triggered by a transient event, then the loop model may be more appropriate. Functional inhibition is not necessary for termination of repetitive bursting in the models presented here. This may have implications for biologic systems where synaptic inhibition is either chronically or acutely diminished. The benefits of neural networks to study synchronous bursting activity are numerous. The specific circuitry is known and can be modified, allowing investigations of the various factors that influence the behavior being studied. In the present studies cessation of bursting could be modeled in a variety of paradigms. In contrast to intact biological systems, neural network models have a relatively limited number of neurons in the system. When larger arrays are employed, computational efficiency can be improved 13

by using simplified models of each neuron. Previous investigations (Kudela et al., 1997, 1999) have shown that these neuronal networks, where neurons are modeled as single compartment components, can behave in a realistic fashion. This is not to discount the important contributions of multiple compartments (e.g. active dendritic properties) to the behavior of each individual neuron. The models chosen here for study were selected because of the ability to model repetitive bursting activity. As mentioned in the methods, the parameters chosen for single neurons are based on the principles of membrane and synaptic physiology. The ratio of inhibitory to excitatory neurons is based on established models (Traub and Miles, 1991). Our ratio of three inhibitory to two excitatory inputs per neuron is a model that provides both background synaptic and excitatory activity as well as bursting under selected conditions. In the simulations reported here repetitive bursting can be produced by either random external action potentials or by external stimulation. The ratio of inhibitory to excitatory inputs is not critical for these experiments since bursting can occur in the absence of functional inhibition. In these models the relative effects of inhibitory versus excitatory inputs can be adjusted by either changing the number of synaptic inputs or by adjusting synaptic weight. For instance, in the model for pure excitatory connections (Fig. 9), the inhibitory inputs are maintained, but making their synaptic weights zero eliminates their influence. The most important intrinsic property that influences production and termination of repetitive bursting is the after-hyperpolarization potential. In the studies reported here we intentionally chose two distinct network models. In the single network model there is constant random excitatory input. In the dual network model, there is no such constant input, instead a feedback loop is modeled. Interestingly, the burst termination produced by external stimuli in the local single network models (Figs. 3 and 4) only resulted in a temporary cessation of bursting. Bursting recurs after several seconds as a result of the background synaptic activity used in this model for activation. In the model with an external feedback loop, there is more regular periodicity to the bursts produced by a single stimulus and, interestingly, external stimulation produces long-lasting cessation of synchronous bursting (Figs. 6 and 9). In the loop model, the timing of the stimulus that

will terminate bursting is determined for the given loop length. Once determined, the window for stimulation after an individual burst is can be very critical. Stimuli that are applied outside of the relatively narrow window fail to terminate bursting permanently (Fig. 8). This is due to the underlying mechanisms of cessation of bursting in this model. The stimulation in network A has to occur at the time when network B is still in relative refractory period, preventing it from fully responding to excitation from network A. The refractory period in network A induced by the stimulus must overlap with the time of arrival of the delayed activity from the previous burst in network B. The exact timing is dependent on the delay in the loop and the length of the relative refractory period in the network, which in turn is dependent on both the intrinsic properties of the neurons (IK(Ca) current) and network connections. In these simulations termination by external stimulus is possible for delays of the feedback no longer than 900 ms. For loops with delay shorter than 800 ms the window for termination is larger, but there are also instances of spontaneous cessation of bursting. This spontaneous cessation occurs when the refractory period from the previous burst overlaps with activity transmitted through the loop in the neurons on either end of the loop. Due to the random connections, the interactions of neurons in the network is highly complex and bursts are not fully synchronized, and after several repetitions neurons involved in the loop may fail to fire. A similar effect can be observed in networks with distributed interneuronal latencies. The bursts in neurons are less synchronized and bursting can be terminated with less precise timing or even stop spontaneously. Indeed, the above effects may explain at least some of the failures in the human applications (Lesser et al., 1999; Karceski et al., 2000), where both spontaneous termination of afterdischarges and a high rate of failures of termination by external stimulus were observed. The experimental designs of these human cortical stimulations could only determine the time after onset of afterdischarges and could not guarantee stimulation during a selected period of the repetitive cycle. In simulated networks one can determine these parameters and produce reproducible results following stimulation. This can provide important insights into the parameters that may be necessary for reliable burst termination in other networks. While the termination of repetitive bursting in these

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models is robust (e.g. not affected by the patterns of internal connections), specific parameters for termination may still need to be modified for different (e.g. larger) networks. In human or animal models, the application of seizure detection algorithms could permit application of the external stimuli at a given period of the repetitive bursting cycle, even recognizing that the periodicity of the bursting is changing as the seizure evolves. Similar to the human studies to date, in the neural network models it does not matter whether the external stimuli are applied shortly after the onset of bursting or after bursting has been established for several seconds. Neither of these experimental situations, however, accounts for changes in synaptic efficacy or receptor desensitization (e.g. GABAA mediated inhibition) that may occur after longer duration seizure activity. In the models presented here the depolarizing pulse has a similar effect on all neurons in the local network. Whether repetitive pulses or pulse trains can terminate the bursting in these networks is being investigated. Functional synaptic inhibition does not appear to be a necessary requirement for termination of synchronous bursting as illustrated here (Fig. 9) and in a previous report (Franaszczuk et al., 1999). This observation is an important one since any application of external stimulation for seizure termination cannot be expected to either selectively augment synaptic inhibition or to relatively increase synaptic inhibition in networks with a predominance of excitatory neurons. The cellular and membrane mechanisms underlying termination of synchronous bursting remain to be fully elucidated. In this model all cells have similar intrinsic properties. It is possible that strong stimulation changes active membrane properties and increases (if only temporarily) the refractoriness of the neurons, interrupting the periodic bursting. As has been recently reported (Franaszczuk et al., 1998) spontaneous seizure termination in humans is accompanied by a change in the pattern of seizure discharges, with a monotonic decline in the frequency of the organized rhythmic activity and a transition to intermittent bursting activity. Time-frequency decomposition of the seizure signal reveals a corresponding increase in signal complexity as the seizure evolves prior to natural termination (Bergey and Franaszczuk, 2001). Conceivably external excitatory stimulation that similarly alters the

natural dynamics of a seizure could result in reduced seizure duration and early seizure termination. Seizure termination in humans and intact preparations may result from a variety of mechanisms as discussed above. At the conclusion of seizures in humans, there is very frequently generalized regional suppression of voltage on the EEG. Depressed EEG voltage can result from diminished activity or desynchronization of existing activity; either could contribute to termination of repetitive seizure activity. In the neuronal network models here the stimuli that are effective in terminating seizures result in diminished synaptic activity (both post-synaptic potentials and triggered action potentials). This dramatically reduced activity effectively prevents the occurrence of synchronized bursting, but is not desynchronization per se. Neural network models provide major advantages in studying the various factors that influence the ability of external stimuli to terminate bursting behavior. In this study, we implement a particular model of stimulation, an external current applied simultaneously to all neurons in a particular network. This may not reflect the type of electric field modulation employed in recent experimental studies (Gluckman et al., 2001). However, it may better relate to stimulation by implanted electrodes in specific brain structures. Combining the information gained from these investigations with sophisticated algorithms for early seizure detection will facilitate the application of these methods to seizure termination in animals and humans. Acknowledgements This research was supported by NIH grant NS 38958 References Av-Ron E., 1994. The role of a transient potassium current in a bursting neuron model. J. Math. Biol. 33, 71–87. Av-Ron E., Parnas H., Segel L.A., 1993. A basic biophysical model for bursting neurons. Biol. Cybern. 69, 87– 95. Ben-Menachem E., 1996. Modern management of epilepsy: Vagus nerve stimulation. Baillieres. Clin. Neurol. 5, 841–848. Bergey G.K., Franaszczuk P.J., 2001. Epileptic seizures are characterized by changing signal complexity. Clin. Neurophysiol. 112, 241–249.

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