- Email: [email protected]
The influence of synaptic connectivity on the pattern of bursting behavior in model pyramidal cells
Neurocomputing 44–46 (2002) 233 – 242
www.elsevier.com/locate/neucom
The in uence of synaptic connectivity on the pattern of bursting behavior in model pyramidal cells Keun-Hang Yang ∗ , Piotr J. Franaszczuk, Gregory K. Bergey Department of Neurology, Johns Hopkins Epilepsy Center, Johns Hopkins University School of Medicine, 600 Wolfe Street, Meyer 2-147, 21287 Baltimore, MD, USA
Abstract Epileptic events observed in humans and animals are associated with synchronized bursting activity in cortical neurons. Recent studies have suggested that a reduced neuron model of synaptically connected neurons can produce repetitive bursting activity. We hypothesize that a reduced model of multicompartment pyramidal neurons can also produce burst activity by changing the synaptic connectivity represented by the synaptic weight and delay. We build a simple pyramidal neuron model of two synaptically connected neurons. Simulations show that a reduced pyramidal neuron model can accurately replicate bursting behavior. The pattern of bursting activity is dependent on the synaptic weight and delay in several locations of excitatory c 2002 Elsevier Science B.V. All rights synaptic connections between neurons in this model. reserved. Keywords: Epileptic events; Bursting activity; Pyramidal neuron
1. Introduction Epileptic seizures observed in humans and animals are associated with synchronized neuronal bursting activity. Recent investigations have suggested that a relatively simple neuron model can produce bursting behavior when the modeled neurons are connected to other similar neurons [6 –8]. Franaszczuk et al. [6] and Kudela et al. [7] suggested that reduced single neuronal models can reproduce a wide range of bursting activity. Kudela et al. [8] studied the spread of synchronous repetitive ;ring in model networks ∗
Corresponding author. Tel.: +1-410-502-8059; fax: +1-410-955-0751. E-mail address: [email protected] (K.-H. Yang).
c 2002 Elsevier Science B.V. All rights reserved. 0925-2312/02/$ - see front matter PII: S 0 9 2 5 - 2 3 1 2 ( 0 2 ) 0 0 4 3 9 - 3
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of locally connected neurons. They found a range of synaptic weights for which the velocity of propagation is consistent with actual measurements of the propagation of epileptiform activity in neocortex [4]. Firing patterns in biophysical neuron models can be described by model parameters [1,2,9]. Av-Ron et al. [2] and Av-Ron [1] suggested that changing certain parameters in a biophysical model may explain bursting activity as well as the ;ring frequency. Kudela et al. [9] suggested that increasing the number of excitatory connections, represented by the synaptic weight, increases the bursting activity in their network model. It is known that various forms of synaptic reorganization in brain tissue may occur in the evolution of epileptic foci in clinical and experimental situations. The aim of this computational study is to investigate the in uence of various parameters of connectivity on the pattern of burst activity in model pyramidal neurons. The spread of activity in synaptically connected pyramidal neurons was investigated using a simpli;ed multicompartmental model. The model simulations were used to ;nd the synaptic weight and delay which generate repetitive bursting behavior of pyramidal neurons in several locally connected regions between neurons. 2. Methodology We have built a reduced pyramidal model using the simulation software GENESIS (http://www.genesis-sim.org/GENESIS/). Three simpli;ed pyramidal neurons are modeled in this study: two neurons synaptically connected as a loop, and a neuron where random input is applied to generate action potentials. The generated action potentials stimulate one of the other neurons which is synaptically connected to the other one as a recurrent loop. There are three types of potential connections between two synaptically connected neurons. These potential connections include synaptic inputs on each soma, synaptic inputs on the main dendrite and branch dendrite, and synaptic inputs on each neuronal branch dendrite of the two neurons (Fig. 1A–C). Each cell is comprised of a soma, a main dendrite, and two branch dendrites, modeled with 15 compartments. The channel equations used in this model are the same as in the Traub et al. multicompartmental CA3 pyramidal cell model [10]. The soma, and the main and branch dendrites have synaptic channels which connect two neurons as a loop. The synaptic connection between neurons is modeled by a synaptic channel, (A) random input
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Fig. 1. Schematic representation of neural connections. There are three types of potential connections between two neurons connected with an excitatory synapse: (A) synaptic inputs on each soma of two neurons; (B) synaptic inputs on each main dendrite of the two neurons; (C) Synaptic inputs on each branch dendrite of two neurons.
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Fig. 2. The patten of bursting behavior when the synaptic inputs are on each soma of two neurons as shown in Fig. 1A. The bursting activity occurs at a range of values of the synaptic delay, 0 –10 ms, and values of the synaptic weight, ¿100 (closed circles). When the synaptic weight is 100 there is no bursting activity for any values of the synaptic delay (open triangles). When the synaptic weight is ¿100 there is no repetitive bursting activity if the synaptic delay is larger than 10 ms; prolonged depolarization occurs (open squares).
Isyn [3]. The synaptic conductance is modeled as an alpha function with the maximum value of 0:5 ns. The synaptic weight represents the overall strength of a connection and the synaptic delay represents all delays between neurons. Simulations were performed for 10 sec: using Genesis version 2.1 on a UNIX operating system.
3. Results Simulations show that a reduced pyramidal neuron model can produce repetitive burst activity which depends upon synaptic parameters represented as the synaptic weight and delay in the synaptic connections between neurons. When the synaptic inputs are on each soma of the two neurons connected with excitatory synapses (Fig. 1A), bursting activity occurs at values of the synaptic weight, ¿ 100, and a range of values of the synaptic delay, 0 –10 ms (closed circles in Fig. 2). When the synaptic weight is decreased to 100 there is no bursting activity for any values of the synaptic delay in these neurons (open triangles in Fig. 2). When the synaptic delay is above 10 ms and the values of the synaptic weight are ¿100, there is no bursting activity; instead prolonged depolarization occurs (open squares in Fig. 2). Fig. 3A–C shows traces of membrane potentials for all neurons with the value of
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Fig. 3. Traces of membrane potentials for all neurons when excitatory synaptic connections are between each soma of neuron 1 and neuron 2 as shown in Fig. 1A: (A) when the synaptic weight and synaptic delay are 100 and 7 ms, respectively, bursting activity does not occur in neuron 1 and neuron 2; (B) when the synaptic weight and synaptic delay are 700 and 7 ms, respectively, repetitive bursting activity occurs in neuron 1 and neuron 2; (C) when the synaptic weight and synaptic delay are 900 and 7 ms, respectively, prolonged depolarization occurs in neuron 1 and neuron 2.
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synaptic delay to be 7 ms and the values of synaptic weights to be 100, 700, or 900. There is no bursting activity in neuron 1 and neuron 2 when the synaptic weight is 100 and the synaptic delay is 7 ms (Fig. 3A). When the synaptic weight is increased to 700 with the same synaptic delay, 7 ms, bursting activity occurs in neuron 1 and neuron 2 (Fig. 3B). When the synaptic weight is increased to 900 with the same synaptic delay, 7 ms, prolonged depolarization occurs in neuron 1 and neuron 2 (Fig. 3C). In order to generate bursts of action potentials from the synaptic inputs on each main or branch dendrite of the two neurons connected with excitatory synapses, the synaptic weight needs to be increased further (compared to somatic inputs). When the values of the synaptic weight are increased from ∼100 to ¿800, bursting activity results from synaptic inputs on each main dendrite of the two neurons connected with excitatory synapses (closed circles in Fig. 4). A range of values of the synaptic delay, 0 –50 ms, is required to generate repetitive bursting activity (closed circles in Fig. 4). When the synaptic inputs are located on each branch dendrite of the two neurons connected with excitatory synapses, bursting activity occurs only if the values of the synaptic weight are increased to ¿8000 (closed circles in Fig. 6). A range of values of the synaptic delay, 0 –50 ms, is required to generate repetitive bursting activity in neurons (closed circles in Fig. 6). Fig. 5A–C shows traces of membrane potentials for all neurons with the value of synaptic delay, 7 ms, and the values of synaptic weights, 800, 1200, or 2000, when the synaptic inputs are on each main dendrite of the two neurons (Fig. 1B).
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Fig. 4. The patten of bursting behavior when the synaptic inputs are on each main dendrite of two neurons as shown in Fig. 1B. The bursting activity occurs at a range of values of the synaptic delay, 0 –50 ms, and values of the synaptic weight, ¿800 (closed circles). When the synaptic weight is 700 or 800, there is no bursting activity for any values of the synaptic delay (open triangles). When the synaptic weight is ¿1400 there is no repetitive bursting activity if the synaptic delay is larger than 3 ms; prolonged depolarization occurs (open squares).
There is no bursting activity in neuron 1 and neuron 2 when the synaptic weight is 800 and the synaptic delay is 7 ms (Fig. 5A). When the synaptic weight is increased to 1200 with the same delay, 7 ms, bursting activity occurs in neuron 1 and neuron 2 (Fig. 5B). When the synaptic weight is increased to 2000 with the same delay, 7 ms, prolonged depolarization occurs in neuron 1 and neuron 2 (Fig. 5C). Fig. 7A and B shows traces of membrane potentials for all neurons when the synaptic inputs are on each branch dendrite of two neurons (Fig. 1C). Fig. 7A shows that there is no bursting activity in neuron 1 and neuron 2 when the synaptic weight is 8400 and the synaptic delay is 7 ms. When the synaptic weight is increased to 10,000 with the same delay, 7 ms, bursting activity occurs in neuron 1 and neuron 2 (Fig. 7B). 4. Conclusions Simulations show that a reduced pyramidal neuron model can accurately replicate bursting behavior. The pattern of bursting activity is dependent upon the synaptic weight and delay in several locations of excitable synaptic connections in a reduced pyramidal neuron model. In order to generate bursting activity, the synaptic weight needs to be increased when the excitatory synaptic inputs are on the main or branch dendrite, instead of when they are on the soma. Modi;cation of connections may occur as a
K.-H. Yang et al. / Neurocomputing 44–46 (2002) 233 – 242
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Fig. 5. Traces of membrane potentials for all neurons when excitatory synaptic connections are between each main dendrite of neuron 1 and neuron 2 as shown in Fig. 1B: (A) when the synaptic weight and synaptic delay are 900 and 7 ms, respectively, bursting activity does not occur in neuron 1 and neuron 2; (B) when the synaptic weight and synaptic delay are 1200 and 7 ms, respectively, bursting activity occurs in neuron 1 and neuron 2; (C) when the synaptic weight and synaptic delay are 2000 and 7 ms, respectively, prolonged depolarization occurs in neuron 1 and neuron 2.
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Fig. 6. The patten of bursting behavior when the synaptic inputs are on each branch dendrite of two neurons as shown in Fig. 1C. The bursting activity occurs at a range of values of the synaptic delay, 0 –50 ms, and values of the synaptic weight, ¿8000 (closed circles). When the synaptic weight is 8000 there is no bursting activity for any values of the synaptic delay (open triangles).
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Fig. 7. Traces of membrane potentials for all neurons when excitatory synaptic connections are between each branch dendrite of neuron 1 and neuron 2 as shown in Fig. 1C: (A) when the synaptic weight and synaptic delay are 8400 and 7 ms, respectively, bursting activity does not occur in neuron 1 and neuron 2; (B) when the synaptic weight and synaptic delay are 10,000 and 7 ms, respectively, bursting activity occurs in neuron 1 and neuron 2.
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result of the epileptic seizures [5]. Therefore, this computational model network can provide insights into the changes that may underlie evolving hyperexcitable states (e.g. epilepsy) in intact systems. Acknowledgements This research was supported by NIH grant NS38958. References [1] E. Av-Ron, The role of a transient potassium current in a bursting neuron model, J. Math. Biol. 33 (1994) 71–87. [2] E. Av-Ron, H. Parnas, L.A. Segel, A basic biophysical model for bursting neurons, Biol. Cybernet. 69 (1993) 87–95. [3] U.S. Bhalla, J.M. Bower, Exploring parameter space in detailed single neurons models: simulations of the mitral and granule cells of the olfactory bulb J. Neurophysiol. 69 (1993) 1948–1965. [4] R.D. Chervin, P.A. Pierce, B.W. Connors, Periodicity and directionality in the propagation of epileptiform discharges across neocortex, J. Neurophysiol. 57 (1988) 12–131. [5] R. Cossart, C. Dinocourt, J.C. Hirsch, A. Merchan-Perez, J. De Felipe, Y. Ben-Ari, M. Esclapez, C. Bernard, Dendritic but not somatic GABAergic inhibition is decreased in experimental epilepsy, Nat. Neurosci. 4 (2001) 52–62. [6] P.J. Franaszczuk, P. Kudela, G.K. Bergey, Realistic modeling of large networks of neurons, Proceedings of Chicago 2000, 2000. [7] P. Kudela, P.J. Franaszczuk, G.K. Bergey, A simple computer model of excitable synaptically connected neurons, Biol. Cybernet. 77 (1997) 71–77. [8] P. Kudela, P.J. Franaszczuk, G.K. Bergey, Model of the propagation of synchronous ;ring in a reduced neuron network, Neurocomputing 26 –27 (1999) 411–418. [9] P. Kudela, P.J. Franaszczuk, G.K. Bergey, Increase in number of synaptic connections increases bursting activity in simulated neural networks, Epilepsia 41 (Suppl. 7) (2000) 6. [10] R.D. Traub, R.K. Wong, R. Miles, H. Michelson, A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances, J. Neurophysiol. 66 (1991) 635–650.
www.elsevier.com/locate/neucom
The in uence of synaptic connectivity on the pattern of bursting behavior in model pyramidal cells Keun-Hang Yang ∗ , Piotr J. Franaszczuk, Gregory K. Bergey Department of Neurology, Johns Hopkins Epilepsy Center, Johns Hopkins University School of Medicine, 600 Wolfe Street, Meyer 2-147, 21287 Baltimore, MD, USA
Abstract Epileptic events observed in humans and animals are associated with synchronized bursting activity in cortical neurons. Recent studies have suggested that a reduced neuron model of synaptically connected neurons can produce repetitive bursting activity. We hypothesize that a reduced model of multicompartment pyramidal neurons can also produce burst activity by changing the synaptic connectivity represented by the synaptic weight and delay. We build a simple pyramidal neuron model of two synaptically connected neurons. Simulations show that a reduced pyramidal neuron model can accurately replicate bursting behavior. The pattern of bursting activity is dependent on the synaptic weight and delay in several locations of excitatory c 2002 Elsevier Science B.V. All rights synaptic connections between neurons in this model. reserved. Keywords: Epileptic events; Bursting activity; Pyramidal neuron
1. Introduction Epileptic seizures observed in humans and animals are associated with synchronized neuronal bursting activity. Recent investigations have suggested that a relatively simple neuron model can produce bursting behavior when the modeled neurons are connected to other similar neurons [6 –8]. Franaszczuk et al. [6] and Kudela et al. [7] suggested that reduced single neuronal models can reproduce a wide range of bursting activity. Kudela et al. [8] studied the spread of synchronous repetitive ;ring in model networks ∗
Corresponding author. Tel.: +1-410-502-8059; fax: +1-410-955-0751. E-mail address: [email protected] (K.-H. Yang).
c 2002 Elsevier Science B.V. All rights reserved. 0925-2312/02/$ - see front matter PII: S 0 9 2 5 - 2 3 1 2 ( 0 2 ) 0 0 4 3 9 - 3
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K.-H. Yang et al. / Neurocomputing 44–46 (2002) 233 – 242
of locally connected neurons. They found a range of synaptic weights for which the velocity of propagation is consistent with actual measurements of the propagation of epileptiform activity in neocortex [4]. Firing patterns in biophysical neuron models can be described by model parameters [1,2,9]. Av-Ron et al. [2] and Av-Ron [1] suggested that changing certain parameters in a biophysical model may explain bursting activity as well as the ;ring frequency. Kudela et al. [9] suggested that increasing the number of excitatory connections, represented by the synaptic weight, increases the bursting activity in their network model. It is known that various forms of synaptic reorganization in brain tissue may occur in the evolution of epileptic foci in clinical and experimental situations. The aim of this computational study is to investigate the in uence of various parameters of connectivity on the pattern of burst activity in model pyramidal neurons. The spread of activity in synaptically connected pyramidal neurons was investigated using a simpli;ed multicompartmental model. The model simulations were used to ;nd the synaptic weight and delay which generate repetitive bursting behavior of pyramidal neurons in several locally connected regions between neurons. 2. Methodology We have built a reduced pyramidal model using the simulation software GENESIS (http://www.genesis-sim.org/GENESIS/). Three simpli;ed pyramidal neurons are modeled in this study: two neurons synaptically connected as a loop, and a neuron where random input is applied to generate action potentials. The generated action potentials stimulate one of the other neurons which is synaptically connected to the other one as a recurrent loop. There are three types of potential connections between two synaptically connected neurons. These potential connections include synaptic inputs on each soma, synaptic inputs on the main dendrite and branch dendrite, and synaptic inputs on each neuronal branch dendrite of the two neurons (Fig. 1A–C). Each cell is comprised of a soma, a main dendrite, and two branch dendrites, modeled with 15 compartments. The channel equations used in this model are the same as in the Traub et al. multicompartmental CA3 pyramidal cell model [10]. The soma, and the main and branch dendrites have synaptic channels which connect two neurons as a loop. The synaptic connection between neurons is modeled by a synaptic channel, (A) random input
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Fig. 1. Schematic representation of neural connections. There are three types of potential connections between two neurons connected with an excitatory synapse: (A) synaptic inputs on each soma of two neurons; (B) synaptic inputs on each main dendrite of the two neurons; (C) Synaptic inputs on each branch dendrite of two neurons.
K.-H. Yang et al. / Neurocomputing 44–46 (2002) 233 – 242
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Fig. 2. The patten of bursting behavior when the synaptic inputs are on each soma of two neurons as shown in Fig. 1A. The bursting activity occurs at a range of values of the synaptic delay, 0 –10 ms, and values of the synaptic weight, ¿100 (closed circles). When the synaptic weight is 100 there is no bursting activity for any values of the synaptic delay (open triangles). When the synaptic weight is ¿100 there is no repetitive bursting activity if the synaptic delay is larger than 10 ms; prolonged depolarization occurs (open squares).
Isyn [3]. The synaptic conductance is modeled as an alpha function with the maximum value of 0:5 ns. The synaptic weight represents the overall strength of a connection and the synaptic delay represents all delays between neurons. Simulations were performed for 10 sec: using Genesis version 2.1 on a UNIX operating system.
3. Results Simulations show that a reduced pyramidal neuron model can produce repetitive burst activity which depends upon synaptic parameters represented as the synaptic weight and delay in the synaptic connections between neurons. When the synaptic inputs are on each soma of the two neurons connected with excitatory synapses (Fig. 1A), bursting activity occurs at values of the synaptic weight, ¿ 100, and a range of values of the synaptic delay, 0 –10 ms (closed circles in Fig. 2). When the synaptic weight is decreased to 100 there is no bursting activity for any values of the synaptic delay in these neurons (open triangles in Fig. 2). When the synaptic delay is above 10 ms and the values of the synaptic weight are ¿100, there is no bursting activity; instead prolonged depolarization occurs (open squares in Fig. 2). Fig. 3A–C shows traces of membrane potentials for all neurons with the value of
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Fig. 3. Traces of membrane potentials for all neurons when excitatory synaptic connections are between each soma of neuron 1 and neuron 2 as shown in Fig. 1A: (A) when the synaptic weight and synaptic delay are 100 and 7 ms, respectively, bursting activity does not occur in neuron 1 and neuron 2; (B) when the synaptic weight and synaptic delay are 700 and 7 ms, respectively, repetitive bursting activity occurs in neuron 1 and neuron 2; (C) when the synaptic weight and synaptic delay are 900 and 7 ms, respectively, prolonged depolarization occurs in neuron 1 and neuron 2.
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synaptic delay to be 7 ms and the values of synaptic weights to be 100, 700, or 900. There is no bursting activity in neuron 1 and neuron 2 when the synaptic weight is 100 and the synaptic delay is 7 ms (Fig. 3A). When the synaptic weight is increased to 700 with the same synaptic delay, 7 ms, bursting activity occurs in neuron 1 and neuron 2 (Fig. 3B). When the synaptic weight is increased to 900 with the same synaptic delay, 7 ms, prolonged depolarization occurs in neuron 1 and neuron 2 (Fig. 3C). In order to generate bursts of action potentials from the synaptic inputs on each main or branch dendrite of the two neurons connected with excitatory synapses, the synaptic weight needs to be increased further (compared to somatic inputs). When the values of the synaptic weight are increased from ∼100 to ¿800, bursting activity results from synaptic inputs on each main dendrite of the two neurons connected with excitatory synapses (closed circles in Fig. 4). A range of values of the synaptic delay, 0 –50 ms, is required to generate repetitive bursting activity (closed circles in Fig. 4). When the synaptic inputs are located on each branch dendrite of the two neurons connected with excitatory synapses, bursting activity occurs only if the values of the synaptic weight are increased to ¿8000 (closed circles in Fig. 6). A range of values of the synaptic delay, 0 –50 ms, is required to generate repetitive bursting activity in neurons (closed circles in Fig. 6). Fig. 5A–C shows traces of membrane potentials for all neurons with the value of synaptic delay, 7 ms, and the values of synaptic weights, 800, 1200, or 2000, when the synaptic inputs are on each main dendrite of the two neurons (Fig. 1B).
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Fig. 4. The patten of bursting behavior when the synaptic inputs are on each main dendrite of two neurons as shown in Fig. 1B. The bursting activity occurs at a range of values of the synaptic delay, 0 –50 ms, and values of the synaptic weight, ¿800 (closed circles). When the synaptic weight is 700 or 800, there is no bursting activity for any values of the synaptic delay (open triangles). When the synaptic weight is ¿1400 there is no repetitive bursting activity if the synaptic delay is larger than 3 ms; prolonged depolarization occurs (open squares).
There is no bursting activity in neuron 1 and neuron 2 when the synaptic weight is 800 and the synaptic delay is 7 ms (Fig. 5A). When the synaptic weight is increased to 1200 with the same delay, 7 ms, bursting activity occurs in neuron 1 and neuron 2 (Fig. 5B). When the synaptic weight is increased to 2000 with the same delay, 7 ms, prolonged depolarization occurs in neuron 1 and neuron 2 (Fig. 5C). Fig. 7A and B shows traces of membrane potentials for all neurons when the synaptic inputs are on each branch dendrite of two neurons (Fig. 1C). Fig. 7A shows that there is no bursting activity in neuron 1 and neuron 2 when the synaptic weight is 8400 and the synaptic delay is 7 ms. When the synaptic weight is increased to 10,000 with the same delay, 7 ms, bursting activity occurs in neuron 1 and neuron 2 (Fig. 7B). 4. Conclusions Simulations show that a reduced pyramidal neuron model can accurately replicate bursting behavior. The pattern of bursting activity is dependent upon the synaptic weight and delay in several locations of excitable synaptic connections in a reduced pyramidal neuron model. In order to generate bursting activity, the synaptic weight needs to be increased when the excitatory synaptic inputs are on the main or branch dendrite, instead of when they are on the soma. Modi;cation of connections may occur as a
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Fig. 5. Traces of membrane potentials for all neurons when excitatory synaptic connections are between each main dendrite of neuron 1 and neuron 2 as shown in Fig. 1B: (A) when the synaptic weight and synaptic delay are 900 and 7 ms, respectively, bursting activity does not occur in neuron 1 and neuron 2; (B) when the synaptic weight and synaptic delay are 1200 and 7 ms, respectively, bursting activity occurs in neuron 1 and neuron 2; (C) when the synaptic weight and synaptic delay are 2000 and 7 ms, respectively, prolonged depolarization occurs in neuron 1 and neuron 2.
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K.-H. Yang et al. / Neurocomputing 44–46 (2002) 233 – 242 (C)
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Fig. 5. (continued)
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Fig. 6. The patten of bursting behavior when the synaptic inputs are on each branch dendrite of two neurons as shown in Fig. 1C. The bursting activity occurs at a range of values of the synaptic delay, 0 –50 ms, and values of the synaptic weight, ¿8000 (closed circles). When the synaptic weight is 8000 there is no bursting activity for any values of the synaptic delay (open triangles).
K.-H. Yang et al. / Neurocomputing 44–46 (2002) 233 – 242
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Fig. 7. Traces of membrane potentials for all neurons when excitatory synaptic connections are between each branch dendrite of neuron 1 and neuron 2 as shown in Fig. 1C: (A) when the synaptic weight and synaptic delay are 8400 and 7 ms, respectively, bursting activity does not occur in neuron 1 and neuron 2; (B) when the synaptic weight and synaptic delay are 10,000 and 7 ms, respectively, bursting activity occurs in neuron 1 and neuron 2.
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result of the epileptic seizures [5]. Therefore, this computational model network can provide insights into the changes that may underlie evolving hyperexcitable states (e.g. epilepsy) in intact systems. Acknowledgements This research was supported by NIH grant NS38958. References [1] E. Av-Ron, The role of a transient potassium current in a bursting neuron model, J. Math. Biol. 33 (1994) 71–87. [2] E. Av-Ron, H. Parnas, L.A. Segel, A basic biophysical model for bursting neurons, Biol. Cybernet. 69 (1993) 87–95. [3] U.S. Bhalla, J.M. Bower, Exploring parameter space in detailed single neurons models: simulations of the mitral and granule cells of the olfactory bulb J. Neurophysiol. 69 (1993) 1948–1965. [4] R.D. Chervin, P.A. Pierce, B.W. Connors, Periodicity and directionality in the propagation of epileptiform discharges across neocortex, J. Neurophysiol. 57 (1988) 12–131. [5] R. Cossart, C. Dinocourt, J.C. Hirsch, A. Merchan-Perez, J. De Felipe, Y. Ben-Ari, M. Esclapez, C. Bernard, Dendritic but not somatic GABAergic inhibition is decreased in experimental epilepsy, Nat. Neurosci. 4 (2001) 52–62. [6] P.J. Franaszczuk, P. Kudela, G.K. Bergey, Realistic modeling of large networks of neurons, Proceedings of Chicago 2000, 2000. [7] P. Kudela, P.J. Franaszczuk, G.K. Bergey, A simple computer model of excitable synaptically connected neurons, Biol. Cybernet. 77 (1997) 71–77. [8] P. Kudela, P.J. Franaszczuk, G.K. Bergey, Model of the propagation of synchronous ;ring in a reduced neuron network, Neurocomputing 26 –27 (1999) 411–418. [9] P. Kudela, P.J. Franaszczuk, G.K. Bergey, Increase in number of synaptic connections increases bursting activity in simulated neural networks, Epilepsia 41 (Suppl. 7) (2000) 6. [10] R.D. Traub, R.K. Wong, R. Miles, H. Michelson, A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances, J. Neurophysiol. 66 (1991) 635–650.