Realistic Models of Neurons and Neuronal Networks

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1 Thoas E. (2002) Realistic odels of neurons and neuronal networks in Encyclopaedia of Cognitive Science, Macillan, Nature Publishing Group (to appear). Realistic Models of Neurons and Neuronal Networks Elizabeth Thoas Research Ctr for Cellular and Molecular Neurobiology Bat L1, Place Delcour Liège, Belgiu integrate and fire odel#rate odel#conductance odels#synaptic odel#ulticopartent odel integrate and fire odel leaky integrate and fire odel rate odel synaptic input to the integrate and fire odel conductance odels kinetic odels of ionic channels synaptic input to conductance odels odels of synaptic odification ulticopartental odels This is an article that describes realistic odels of neurons. The odels range in coplexity fro the perfect integrate and fire odel to conductance odels in which the ionic currents are coputed fro the kinetics of the underlying gating particles. 1. Introduction This section outlines the construction of neuronal odels which describe in ore detail their intrinsic properties as well as their interactions with neighbouring neurons. It starts with the integrate and fire neuron as a odel on the less detailed end of the spectru, while we finish with odels in which the ionic currents eerge fro the kinetics of the underlying gating particles. The coplexity of the odel chosen for a study would depend largely on the type of question being investigated. For exaple, one cannot fully investigate the effects of any drugs on a neuronal network without the use of the Hodgkin Huxley odel. The reason being that in any cases, the effect of the drug is known in soe detail e.g. benzodiazepines ostly effect the decay of γ- Ainobutyric acid A (GABA A ) ediated postsynaptic potentials and not the axiu conductance (Otis and Mody, 1992). On the other hand, if one is investigating auditory processing, there is already uch which is unknown about the anner in which inforation is encoded in a siple network that constructing a coplicated conductance odel only serves to confound the proble. The extra details ay also be irrelevant to the questions being addressed. In describing various realistic neuronal odels starting out with the siplest, I will begin with the integrate and fire odel. I will then ove on to the leaky integrate 1

2 and fire odel, the rate odel of a neuron and finally conductance odels. The conductance odels can be constructed either with acroscopic descriptions of ionic currents or with ore detailed odels that outline the kinetics of each of the particles gating the channel. A study of network interactions requires odels of synaptic activity. I will therefore describe two coonly used synaptic odels odels with prefixed functions of postsynaptic activity and kinetic odels. Finally since phenoena such as learning and eory require the investigation of neuronal plasticity, I present biological odels of both short and long ter plasticity. 2. Integrate and Fire Model The integrate and fire odel of neurons does not include any detailed inforation on the intrinsic properties of the neuron. Input to the neuron is integrated and a spike or action potential is generated once a fixed threshold is reached. The spiking property of the neurons allows for a study of the role of factors such as spike tiing and spike synchrony in network activity. Soe recent connectionist odels incorporating the spiking activity of neurons have been found to yield certain advantages over a network of non-spiking neurons (Sougné, 1998). The integrate and fire neuronal odel is described by the following type of equation: C dv dt = I(t) (1) C is the capacitance of the neuron, V is the ebrane potential and I(t) is the suated input current due to synaptic input fro neighbouring neurons as well as any external current source such as an electrode. The neuron fires a fixed action potential when threshold is reached. The ebrane potential is then set to a prefixed value after the spike. Iplicit in such a odel is the idea that the tiing of a spike and not its for, is iportant in carrying inforation concerning the syste. The siplest way of representing a spike fro the neuron is a siple delta function δ(t-τ k ) for a neuron which fires at tie τ k. While this is the siplest representation that can be used, various coplications can be added to the odel to ore realistically characterise the neuronal spike and ore iportantly its activity iediately afterwards. Perhaps the ost iportant of these is the refractory period iediately following an action potential. During this period no stiulus is able to elicit a second action potential fro the neuron. The refractory period is due to the fact that the sodiu channels largely responsible for generating a spike are in an inactivated state. The presence of this constraint places an upper liit on the frequencies at which a neuron ay fire an action potential. If the refractory period is t ref, the axiu firing frequency of the neuron, f ax is 1/t ref. The neuronal odel ay also include a relative refractory period iediately following the refractory period during which the threshold for evoking a spike is increased. 2

3 3. Leaky integrate and fire odel These odels add a new level of coplexity in the behavior of a neuronal odel by the inclusion of a leak current. The neuronal ebrane in this instance can be represented by the equivalent circuit in Figure 1. The current flowing across the resistance of the circuit represents the leak current. The total current I flowing across the neuronal ebrane is the su of the capacitance current I C and leak current I l dv dt I = C + V R (2) The siple integrate and fire odel would linearly integrate two teporally separated inputs provide their su does not exceed threshold. The presence of a leak however introduces a ore realistic behavior in which the occurrence of an input is gradually forgotten over tie. The earlier occurring input therefore akes a less iportant contribution to the suated neuronal response at a latter tie. In the presence of a leak ter, the ebrane potential in the absence of any input current will gradually decay to its initial value. In response to a constant current pulse I, the leaky integrate and fire odel would obtain the steady state value V ( )=I R. The value of the ebrane potential before this steady state is reached however is given by V ( t) = I R (1 e t / τ ) (3) Where τ =R C and is the ebrane tie constant representing the tie taken for V to reach 63% of its final value. (approxiate position of Figure 1) 4. Rate Model In any odels of large networks, the neuronal output is not represented in the for of individual spikes. Instead the inforation taken into account is the firing rate of a neuron. In this case, the neuronal output is coputed fro a sigoidal function relating ebrane potential to the neuronal firing rate. The instantaneous firing rate z i (t) of a neuron is usually obtained fro the neuronal ebrane potential V (t) with the use of a sigoid function z ( ) = ax i t b( V( t ) θ ) 1+ e z (4) The use of this function allows for two factors contributing to nonlinearities in the neuronal output. Due to the refractory period, there is a axial rate at which the neurons can fire z ax. The second factor is a threshold to reflect the fact that a iniu 3

4 excitation is required for the neuron to begin firing. The factors b and θ contribute to setting the threshold of excitation and also the rate at which the firing increases with ebrane potential. The use of a rate odel of course does not allow for the use of inforation that ight be represented in the tiing of neuronal firings or the correlation of firing ties between neurons. The question of whether a rate code or teporal code is used for inforation coding in the nervous syste has been a very active one for the last few years. Aong the investigators who have presented evidence for the representation of stiulus inforation in the teporal odulation of the neuronal spike train have been Bialek et al (1991) who worked in the visual fly syste and Richond et al (1987) who investigated the inferior teporal cortex. The stance that spike tiing in neuronal firing is not very iportant has been taken by investigators such as Shadlen and Newsoe (1998) and Thoas et al (2000). The latter investigators present evidence to deonstrate that a good level of category inforation can be represented in the inferior teporal cortex without inforation concerning spike tiing. 5. Synaptic input to the integrate and fire neuron When present in a network, the neurons receive input fro the surrounding neurons. At a given tie step all the input fro the surrounding neurons are sued linearly to give the resulting activity V i of a neuron i C dvi dt Vi = + i gi ( t k ) j( t k )[ Vi Veq ] j j R ω τ δ τ (5) j k The double suation is to take into account not only the input fro all the presynaptic neurons j, but also to su over each incidence k of a presynaptic neuron firing. The ter ω ij is the sign and the strength of connections between neurons i and j. The function g ij (tt 0 ) usually takes the for of an alpha function g ij ( t t0) ( t t0) = e τ ( t t0) / τ (6) Where t o is the tie of transitter release. The function g ij (t-t 0 ) decays with a tie constant τ after reaching a peak. The values of τ are chosen to reflect the various fast and slow excitatory and inhibitory synapses seen in the nervous syste. In any realistic network odels these include the fast inhibitory synapse GABA A, the slow inhibitory synapse GABA B, fast excitation ediated by aino-3-hydroxy-5-ethyl-4-isoxazoleproprionic acid (AMPA) receptors and a slow excitation ediated by N-ethyl-Daspartate (NMDA) receptors. Each synapse also has its own characteristic value of V eq where (V-V eq ) represents the driving force on each synaptic current. 4

5 6.Conductance odels Many studies in Neuroscience involve an investigation of the neuronal ionic channels. The effects of various drugs on neurons are understood in ters of their actions on the ionic channels. The fraework priarily used to describe these channels is the Hodgkin Huxley odel (1952). One of the principle advantages of using this odel is the availability of a large nuber of the required paraeters fro the experiental literature. This is because any of the experiental investigations theselves are carried out using the Hodgkin Huxley fraework. Soe of the disadvantages of using the odel is the large nuber of paraeters required to characterize the syste and the long coputer run ties required to treat such unwieldy systes. Ionic channels in the Hodgkin Huxley odel are represented as resistances (or conductances) arranged in parallel (Figure 2). The equivalent circuit for such a syste is an elaboration of what has already been presented in Figure 1. In this case additional resistances are added to siulate the other currents in the neuronal ebrane in addition to the leak current. The circuit in Fig 2 represents the currents that are typically represented in odels of thalaocortical (TC) cells (Lytton and Thoas, 1997; Thoas and Lytton, 1997; Thoas and Grisar, 2000). The equation which describes such a circuit is the following C dv dt = I T I h I Na I K I l I GABAA I GABAB (7) The currents I T, I h, I Na, I K and I l are intrinsic currents while I GABAA and I GABAB are synaptic currents. The values of these ionic currents are obtained fro Oh s law I = gg ( V Eeq ) (8) E eq is the reversal potential of the channel and g is the axiu conductance of a channel. Both these values are fixed. The value of g, the channel conductance, on the other hand is frequently dependent on things such as the ebrane potential, tie, or the internal calciu concentration and has to be generally coputed for each channel at each tie step. The leak current is usually an exception to this. In the case of the intrinsic channels in which g changes, the value of g depends on the state of the gating particles and h. While is an activation gate, h is an inactivation gate. If N is the nuber of activation gates g = N h (9) The state of the gating particles ( or h) is a function of ebrane potential as well as tie. At any tie t, its value can be coputed fro the equation 5

6 gate t ( )exp[ t gate gate gate ] t 1 (10) τ = gate The values gate α and τ gate depend on ebrane potential. In the case of any channels, their values as a function of ebrane potential is directly available fro experiental literature. In other cases they have to be coputed fro α and β, the rate constants for the reaction between open and closed states for gating particles α gate open gate closed β (11) In this case gate τ gate α( V ) ( V ) = α( V ) + β ( V ) 1 ( V ) = α( V ) + β ( V ) (12) (13) As indicated by the nae, the value gate deterines the agnitude of gate opening when the ebrane potential is held at a particular value for tie infinity. The value of τ gate deterines the aount of tie required for the gate opening. (Approxiate position of figure 2) 7. Kinetic odels of ionic channels The Hodgkin and Huxley representation of ionic currents is one at a acroscopic level. Many odels also portray the currents at a ore detailed level by describing the ionic channel based on the state transitions of each of the gating particles that constitute the channel. The value of an ionic current therefore eerges fro the underlying dynaics of the syste. There are two key assuptions that underlie classical kinetic theory which is used to describe the state transitions of the gating particles 1) Gating is a Markov process i.e. the rate constants of the transition of a gating particle fro one state to the other (eg open to closed) is independent of the previous history of the syste. 2) The transitions can be described by first order differential equations with a single tie constant. A construction of such odels can be illustrated by the odel of a potassiu (K) channel previously described by Hodgkin and Huxley. The channel is ade up of 4 n- gates. Each of these gates can be in either an open (1) or closed (0) state. The channel is only conducting if all four n gates are in an open state (1111). Starting fro the 6

7 condition where all the gates are closed (0000), the syste can therefore go through 14 interediate closed states before finally becoing open (1111) (Figure 3). By using the assuption that the four gates are independent and kinetically identical, the web of equations that lead fro the non-conducting to conducting state of the K ionic channel can be very uch reduced. For exaple all states with the sae nuber of closed gates are kinetically identical and can therefore be treated together. Using such assuptions the entire tree of reactions can be collapsed to the transitions between 5 ajor states, with state 4 being the only conducting state. The values 4α, 3α, 2α, α, are the rate constants for all the forward reactions leading to state 4, while 4β, 3β, 2β and β are the rate constants for the backward reactions leading fro open state 4 to the closed state 0. (approxiate position of figure 3) 8. Synaptic input to conductance odels Synaptic input in any conductance odels is represented in the for of the alpha function introduced in Section 5. As entioned earlier, one of the probles with the use of this odel is the fact that the tie of occurrence of each spike has to be stored in a queue and their corresponding exponentials have to be calculated for each tie step. Another criticis of these odels is that while they iitate the observed tie course of postsynaptic potentials in electrophysiological recordings, they are not based on any of the underlying echaniss in the process. These odels also do not naturally provide for a saturation that could occur at the synapse. A kinetic odel of receptor binding described by Destexhe et al (1994a) is able to resolve any of these probles. In a two state kinetic odel, neurotransitter olecules T, bind to the postsynaptic receptor R according to the first order schee α R + T TR* β (14) If r is the fraction of bound receptors, then dr dt = α[ T ](1 r) βr (15) If [T] occurs as a pulse, the above equation can be solved to obtain the following expression for the value of r during the pulse (t 0 <t<t 1 ) and for the period following the pulse (t>t 1 ). During the pulse ( t t0) r( t to) = r + ( r( t 0) r )exp[ ] τ r After the pulse ( t 0 t < t1) < (16) 7

8 r t t ) = r( t )exp[ β ( t )] t > t ) (17) ( 1 1 t1 ( 1 Where both r and τ r are functions of the rate constants α and β of equation 14. The value of r is then used to calculate the synaptic current in the sae anner as an intrinsic current. I syn ( t) = g synr( t)[ Vsyn( t) Esyn] (18) While these two state odels capture any of the iportant characteristics of postsynaptic potentials, a larger nuber of states can be used to ore accurately odel the (Destexhe et al, 1994b). 9. Models of synaptic odification The efficacy of connections between the neurons of the nervous syste has been found to be a paraeter which is not constant but fluctuates based on the dynaics of the network. These changes can be both long and short ter and can cause synaptic efficacy to increase or decrease. While the short ter changes last on the order of seconds to inutes, the long ter changes can persist for an hour or ore. A odel for short ter synaptic facilitation and depression has been provided by Abbott et al (1997). In this odel, the aplitude of neuronal response K 1 is adjusted by a factor A(t i ). If R is the response of a neuron to a train of spikes R(t) = Σ A(t i )K 1 (t-t i ) (19) If there was an isolated spike and previous spikes did not have an influence on the postsynaptic response, A would have the value 1. As a result of a presynaptic spike however, A would either increase in the case of facilitation or decrease in the case of depression. This could either take place in an additive or ultiplicative fashion. The value of A would change according to A fa in the ultiplicative case or A A+(f-1) in the additive case. A value of f>1 would correspond to facilitation while f<1 would lead to depression. In order to produce biologically realistic behavior, Abbott recoends the usage of a ultiplicative expression for depression (f<1) and an additive description for facilitation (f>1). Another odel of short ter depression can be found by Markra et al (1998). This odel ais to explain the frequency dependence of synaptic transission. Both Markra and Abbott have suggested that these changes in synaptic strength have iplications for the sort of encoding used by the nervous syste. A rate code is favored when depressions is slow and a teporal code when depressions is fast. A neuron which was initially sensitive to the firing rate of a rapidly firing presynaptic terinal, as a result of synaptic depression, now reaches a steady state which is no longer a function of presynaptic firing rate. Several depressed EPSP s however are generated before this steady state is reached. During this period, the neuron shows an increased sensitivity to 8

9 teporal coherence in the presynaptic input (Abbott et al, 1997, Tsodyks and Markra, 1997). The long ter changes observed in the strengths of synaptic connectivity are long ter potentiation and long ter depression. The experiental paradig usually used to produce LTP is a train of high frequency stiulation while LTD is produced by a train of low frequency stiuli. LTP has found to be ediated by both NMDA as well as non- NMDA synapses. Long ter potentiation ediated by NMDA synapses generally obey Hebbian rules for induction. The Hebbian rule was proposed by Donald Hebb as a echanis for learning. He proposed that the concurrent excitation of pre and postsynaptic eleents would result in an increase of synaptic efficacy. In the NMDA synapse the requireent for concurrence arises because the postsynaptic terinal has to be depolarized in order to reove a Mg 2+ block. The presynaptic terinal has to be depolarized for the release of neurotransitter. Realistic odels of the NMDA synapse ust therefore take into account the concentration of Mg 2+. The following is an exaple of an equation for an NMDA ediated current in a odel of working eory in the cortical network (Wang, 1999) I NMDA g NMDA( V VE ) = (20) kv (1 + [ Mg2+ ] e ) More recent experiental work has revealed ore details on how the relative tiing of pre and postsynaptic firing can change the sign as well as the strength of synaptic odification. If the presynaptic action potential precedes postsynaptic firing by no ore than about 50s, then an increase in synaptic strength takes place. The saller the teporal separation between the tiing of the presynaptic firing and the postsynaptic action potential, the larger the increase in synaptic strength. On the other hand, presynaptic action potentials that follow postsynaptic spikes lead to a depression in synaptic strength. Models of this sort of spike tiing dependent plasticity have been developed by Song et al (2000). 10. Multicopartental odels The highly branched dendritic trees of a neuron confer a highly coplex spatial structures to the neuron. Neuronal odels which attept to understand the role of these structures ake use of ulticopartents (figure 4). The copartental approach to study the flow of ionic currents in dendritic trees, was pioneered by Rall (1964). Rall had odeled the dendritic tree with the use of cable equations. These equations could be analytically solved under extreely liited circustances. This approach becae difficult and soeties ipossible when ost realistic dendritic structures were taken into account. Also probleatic are ebrane properties which are voltage dependent and which generate action potentials. Rall had pointed out that in such cases, copartental odels should be used. A good description of such odels can be found in Segev and Burke (1999). In this instance each neuron is ade up of a nuber of copartents. Each copartent is considered isopotential and its properties to be unifor. For such odels, the property of each copartent is not only the consequence 9

10 of currents intrinsic to the copartent but also depend on currents which enter and leave fro adjacent copartents. The ebrane potential for a neuronal copartent k with adjacent copartents k+1 and k-1 is then coputed as follows c dv dt k + I ion V = r k 1 V k 1, k k Vk V r k, k + 1 k + 1 (21) While in any instances, unicopartent odels are sufficient to generate the activity of interest, in soe situations, they fail. An exaple of this is a study described by Abbott and Marder (1998). They deonstrate how in soe bursting cells in the soatogastric ganglion of certain inverterbrates, it is iportant to isolate the priary neurite fro the spike initiation zone on the axon. If this is not done, the slow oscillating potentials generated in the priary neurite can prevent the deinactivation of the fast Na+ conductance which terinates the action potential in the spike initiation zone. Once these two areas are electrotonically separated, the proble is resolved. (Approxiate position of figure 4) References Abbott LF, Varela JA, Sen K, Nelson SB (1997) Synaptic depression and cortical gain control. Science 275: Abbott L and Marder E (1998) Modeling sall networks. In: Koch C and Segev I (eds) Methods in Neuronal Modeling, pp 372. Cabridge, Massachusetts: MIT Press. Bialek W, Rieke F, de Ruyter van Stevenick RR, and Warland D (1991) Reading a neural code. Science 252: Destexhe A, Mainen Z and Sejnowski T (1994b) An efficient ethod for coputing synaptic conductances based on a kinetic odel of receptor binding. Neural Cop 6: Destexhe A, Mainen Z and Sejnowski T (1994b) Synthesis of odels for excitable ebranes, synaptic transission and neuroodulation using a coon kinetic foralis. J Cop Neurosci 1: Hille (1991) Ionic Channels of Excitable Mebranes, pp 487. Sinauer Associates Inc. Massachusetts. Hodgkin AL, and Huxley AF (1952) A quantitative description of ebrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond.). 117: Lytton W and Thoas E (1997) Modeling thalaocortical oscillations. In Cerebral Cortex. Volue 13. Models of Cortical Circuitry. Philip S. Ulinski and Edward G. Jones (Editors), Plenu Press, New York. 10

11 Markra H, Pikus D, Gupta A and Tsodyks M (1998) Potential for ultiple echaniss and algoriths for synaptic plasticity at single synapses. Neuropharacology 37: Otis T and Mody I (1992) Modulation of decay kinetics and frequency of GABA A receptor-ediated spontaneous inhibitory postsynaptic currents in hippocapal neurons. Neuroscience 49: Rall W (1964) Theoretical significance of dendritic trees for neuronal input-output relations. In: Reiss RF (ed) Neural theory and odeling, ed, pp Stanford, CA: Stanford University Press. Richond BJ, Optican L, Podell M and Spitzer H (1987) Teporal encoding of twodiensional patterns by single units in priate inferior teporal cortex. I Response characteristics. Journal of Neurophysiology 57: Segev I and Burke R (1999) Copartental odels of coplex neurons. In: Koch C and Segev I (eds) Methods in Neuronal Modeling, pp Cabridge: MIT Press. Shadlen MN and Newsoe W (1998) The variable discharge of cortical neurons: Iplications for connectivity, coputation and inforation coding. Journal of Neuroscience 18: Sougné J (1998) Connectionis and the proble of ultiple instantiation. Trends in Cognitive Sciences 2: Song S, Miller K and Abbott LF (2000) Copetitive Hebbian learning through spiketiing dependent synaptic plasticity. Nature Neuroscience 3: Thoas E and Lytton W (1997) Coputer odel of antiepileptic effects ediated by alterations in GABA A ediated inhibition. NeuroReport 9(4): Thoas E and Grisar T (2000) Increased synchrony with increase of a low threshold calciu conductance in a odel thalaic network: a phase shift echanis. Neural Coputation 12: Thoas E., Van Hulle M. & Vogels R. (2000). Encoding of categories by non-category specific neurons in the inferior teporal cortex. Journal of Cognitive Neuroscience 13: Tsodyks MV and Markra H (1997) The neural code between neocortical pyraidal neurons depends on neurotransitter release probability. Proc Natl Acad Sci 94: Wang XJ (1999) Synaptic basis of cortical persistent activity: the iportance of NMDA receptors to working eory. J. Neurosci 19: Glossary Integrate and fire neuron odel neuronal odel in which neuron integrates all input and fires when threshold is reached. Leaky integrate and fire neuron odel neuronal odel in which the neuron gradually forgets earlier inputs due to the presence of a leak current. Rate odel non-spiking odel of neuron in which firing rate of neuron is usually coputed fro a sigoid function of neuronal ebrane potential 11

12 Kinetic odels of channels Models of intrinsic neuronal or synaptic channels in which the channel currents are coputed fro the kinetics of the particles which constitute the channel. Conductance odel neuronal odel in which the various ionic channels of a neuron are represented as the resistors in an equivalent circuit Equivalent circuit odel of neuronal activity in ters of electrical coponents such as resistors and capacitors Synaptic plasticity alterations in the efficacy of a synapse. These changes ay be positive or negative and they can be of both short and long ter durations. Further reading Johnston D and Wu S (1995) Foundations of cellular neurophysiology. Massachusetts, The MIT Press. Koch C (1999) Biophysics of Coputation. Oxford, Oxford University Press. Koch C and Segev I (1999) Methods in Neuronal Modeling. Massachusetts, The MIT Press. Hille B (1991) Ionic Channels of Excitable Mebranes. Sinauer Associates Inc. Massachusetts Bower J (1999) Coputational Neuroscience: Trends in Research Elsevier. De Schutter E (2000) Coputational Neuroscience: Realistic Modeling for Experientalists. CRC Press. Traub RD and Miles R (1991) Neuronal Networks of the Hippocapus. Cabridge University Press: Cabridge, United Kingdo. Destexhe A and Sejnowski T (2001) Thalaocortical Asseblies. Oxford University Press. Oxford, United Kingdo. Figure Legends Figure 1 Equivalent circuit for odel of leaky integrate and fire odel neuron 12

13 Figure 2 Equivalent circuit for conductance odel of thalaocortical neuron Figure 3 a) The K channel consisting of 4 n particles can go fro the closed state (0000) to the only open state (1111) via 14 interediate states which are all closed. b) By luping together states which are kinetically identical, the coplex state diagra in a) can be reduced to 5 states (Hille, 1991). Figure 4 Equivalent circuit for ulticopartent odel (Segev and Burke, 1999) 13

14 k-1 k k+1 V k-1 V k V k+1 i k-1,k I ion r i k,k+1 c i Figure 4. Equivalent circuit for ulticopartent odel (Segev and Burke, 1999). 14

15 A B α 2α 3α 4α β 2β 3β 4β Figure 3 a) The K channel consisting of 4 n particles can go fro the closed state (0000) to the only open state (1111) via 14 interediate states which are all closed. b) By luping together states which are kinetically identical, the coplex state diagra in a) can be reduced to 5 states (Hille, 1991). 15

16 inside g K g Na g L g T V C E K E L E T E Na outside Figure 2. Equivalent circuit for conductance odel of thalaocortical neuron. 16

17 I inside I l I c V R C outside Figure 1. Equivalent circuit for odel of leaky integrate and fire odel neuron. 17