A simple computer model of excitable synaptically ... - CiteSeerX
A simple computer model of excitable synaptically connected neurons Pawel Kudela
a;1
Piotr J. Franaszczuk
Gregory K. Bergey
b;2
b;c;2
Laboratory of Medical Physics, Institute of Experimental Physics, Warsaw University, Warsaw (Poland) b Maryland Epilepsy Center, Department of Neurology, University of Maryland School of Medicine and Medical Center, 22 South Greene Street,Baltimore, MD 21201 (USA) c Department of Physiology, University of Maryland School of Medicine, Baltimore, MD 21201 (USA) a
Abstract The space-lumped two variable neuron model is studied. The extension of the neural model by adding a simple synaptic current allows to demonstrate neural interactions. The production of synchronous burst activity in this simple two neuron excitatory loop is modeled, including the in uence of random background excitatory input. The ability of the neuron model to spatially and temporally integrate inputs is shown. Two refractory periods after stimuli were identi ed and their role in burst cessation is shown. Our ndings show that simple neural units without long-lasting membrane processes are capable of generating long lasting patterns of activity. The results of simulation of simple background activity suggest that an increase in background activity tends to cause decreased activity of the network. This phenomenon as well as existence of two refractory periods allows for burst cessation without inhibition in this simple model.
1 Introduction When modeling neural networks, we are usually interested in models that are as simple as possible with regard to computational complexity. On the other hand we would like the models to exhibit certain physiological features. The 1 2
Supported by KBN grant 3P401-00407 Supported by NIH grant NS 33732-01
Preprint submitted to Biological Cybernetics
April 1997
simple spin- ip model (Hop eld and Tank 1986) is easy to implement, but has rather limited physiological value. The choice of model is often a compromise between computing ability and the complexity of the phenomena that the system is intended to simulate. There are many neuronal models derived from the classical Hodgkin-Huxley (1952) model. Some of them, (e.g. Yamada et al. 1989; Chay and Lee 1990), focus on detailed quantitative modeling of ion channels dynamics. Others (e.g. FitzHugh 1961; Rinzel 1985; Av-Ron et al. 1991) concentrate on general, qualitative neuronal properties, such as the generation of individual or repetitive action potentials. Models of neurons from the rst group are more complex and were introduced to model the electrical activity of the individual type of cells. Neuron models belonging to the second group are usually much easier to implement and are simpler to analyze, but are missing some of the physiological features. Our goal is to implement a relatively simple neuron model, with certain physiological membrane features, and to investigate how the modeled neuron behaves when connected to other similar neurons. We will show that, despite their simplicity, such connected neurons exhibit various phenomena observed in real neural networks. It should be noted that this kind of reduced neuron model (Rinzel 1985; Av-Ron et al. 1991) has been studied in the context of the single neuron. This approach limits analysis of the characteristics of the model and might hide some neuronal properties. For example, Rinzel (1985) discusses the phenomena of threshold and repetitive ring in response to a single brief current pulse. In real systems, synaptically connected neurons excite or inhibit each other. Our goal is to study the behavior of a modeled neuron in conditions similar to physiological reality. As a starting point we took the Rinzel's (1985) space-lumped two-variable model. Next we modeled the synaptic input in this model. We examined the ability of the neuron to spatially and temporally integrate inputs. We also analyzed the neuronal output properties for various input stimuli. Then, we added the simulated random background activity to input, and connected two neurons into the feedback loop. Finally, we examined di erent patterns of neural activity, especially burst onset and cessation in the simple neuron loop.
2 Model description 2.1 Model of neuron
We used Rinzel's model of a single neuron as modi ed by Av-Ron et al. (1991). This choice is justi ed by the easier analytical tractability of this model, and its ability to generate di erent types of potentials. The model is based on a simpli cation of the Hodgkin-Huxley model (1952) of excitable membrane (H2
H). Here, following the observation of FitzHugh (1961) that the sum of two H-H variables n and h during an action potential is constant, one common variable is introduced. The second simpli cation in this model is an assumption that the third "fast" Hodgkin-Huxley m variable is in a quasi-steady state m with respect to changing the value of the membrane potential. This approach simpli es the model by reducing the number of variables, while not signi cantly altering the model dynamics. The Av Ron's model is composed of two ordinary di erential equations. The rst equation describes the rate of change of voltage with respect to time as proportional to the sum of three currents owing through the membrane: an inward sodium current INa, an outward potassium current IK , and a leak current IL. Each ion current Ii , is described by a maximal conductance gi and a driving force proportional to (V ? Vi). Formulas de ning currents INa and IK include activation-deactivation terms. The second equation describes the rate of change with respect to time of the recovery component W , a linear combination of two Hodgkin-Huxley variables n and h. W is called the recovery variable because it is responsible for the conclusion of the action potential and return to resting membrane potential. A brief description of the model follows while a more detailed discussion of this neuron model can be found in Av Ron et al. (1991). Denoting membrane potential as V and Rinzel's "recovery" variable as W the basic set of equations is as follows:
Cm dV=dt = I ? gNa mmp 1 (V )(1 ? W )(V ? VNa ) ? gK (W=s)wp(V ? VK ) ? gL(V ? VL)
(1) (2)
dW=dt = (W1(V ) ? W )= (V )
(3)
and
where the parameters mp = 3, wp = 4, and s = 1:3 and W1(V ) and m1(V ) are steady state conductance functions de ned by:
W1(V ) = (1 + exp [?2a W (V ? V =W ])?
(4)
m1(V ) = (1 + exp [?2a m (V ? V =m ])?
(5)
(
(
)
)
( ) 1 2
( ) 1 2
1
1
and
(V ) = ( exp [a W (V ? V =W )] + exp [?a W (V ? V =W )])? (
( ) 1 2
)
(
)
( ) 1 2
1
3
(6) (7)
where is the relaxation time function for recovery variable W . Terms 4 and 5 have sigmoid curve shapes, with the half maximal voltage value of V = and the slope of the curve controlled by a. All the parameters a W , a m , V =W , V =m , and g are chosen to t the experimental data (see Appendix). This model exhibits two types of qualitative behavior. For a short suprathreshold stimulus the model will generate a single action potential. The current I in Eq.1 represents external input stimuli. In our simulations this accounts for input from other neurons. There are other input and parameter values, for which the model yields oscillatory behaviors (e.g. limit cycle oscillations, AvRon et al. 1991). 1 2
(
)
(
)
( ) 1 2
( ) 1 2
Each neuron has a probability of generating a postsynaptic potential at each simulated time interval, independent of synaptic inputs. This simulates spikes arriving from other neurons not modeled, and will be referred to as background activity. The above de nition implies that generation of background activity is a random Poisson process. 2.2 Model of the synaptic connection
Our modeling of neuron connections is based upon the assumption that the neuron integrates inputs from other neurons that generate postsynaptic potentials (PSPs). Inputs can have di erent strengths modeled by synaptic weight and di erent polarity: positive for excitatory inputs, or negative for inhibitory inputs. All delays in transmission of an AP between ring in one neuron and generation of a PSPs in the next neuron are modeled by a single model parameter, the synaptic delay. The synaptic contribution by an external current Isyn is de ned by:
X wj gj(t) (V ? Esyn)
(8)
XN (exp((t ? t)= d) ? exp((ti ? t)= o))
(9)
Isyn = and
g(t) =
Nsyn j =1
i=1
where d and o are respectively, decay and onset parameters, i denotes the time of arrival of i-th (after transmission and synaptic delay); Nsyn is the number of input synapses; and wj stands for the synaptic weight, positive for excitatory and negative for inhibitory connections; Esyn is the synaptic reversal potential. This form of synaptic conductance allows for adjustment of the time course of evoked PSPs by changing values of d and o . Summation 4
over time (Eq.9) is limited to N most recent APs, for which t ? ti is within a preset interval. This interval depends on the decay parameter d . For a d ranging between 2 and 3 milliseconds we limit the summation interval to a range of 15 - 20 milliseconds. For each red AP a synaptic current is added to the postsynaptic neuronal equation after preset synaptic delay. Each neuron can have multiple input synapses and di erent neurons can have di erent number of synapses. Output from each neuron can be sent to several other neurons. Each synaptic connection can have a di erent weight and delay. In this way multiple neuron con gurations can be simulated. In this paper we describe only the simplest interactions between two neurons. The dynamics of simulated neurons is illustrated by the voltage potential representation (plot shows a value V for each modeled neuron in time).
2.3 Computer algorithms and codes
The model equation, (Eq.1 and 3) is integrated using an explicit rst-order Euler scheme to update the neuron membrane voltage, according to the following formula:
V (t + t) = V (t) + (dV (t)=dt) t
(10)
Integration step t is set to 10 s. We choose to use this simple scheme to speed up computation with minimal loss of precision. This is checked by using Runge-Kutta method for integration, and the results of simulation are not signi cantly changed. All simulation programs were written in ANSI C, compiled and run on both the IBM RS6000 workstation running AIX operating system and the PC DOS computers. The main program accepts input parameters from a disk le and writes output data to a disk le. The input data de nes both the neuron model parameters and interneuronal connections. For this simulation, separate programs for generating input parameters and for displaying output data were developed. This allowed the main simulation to be computed on a powerful UNIX workstation, while design of the network and analysis of output could be done interactively on PC computers. Supporting procedures were developed to create input les for the simulation module and to display di erent kinds of output data. 5
3 Results of simulations
3.1 Temporal and spatial summation property of the model
First we examined the ability of our implementation of the neuron model to reproduce basic physiological phenomena. The model can provide an AP in response to a superthreshold stimulus. For a standard input stimulus above in which an AP appears, we determined the threshold value for synaptic weight. We then examined the summation properties of our model with two or more synaptic excitatory stimuli, from di erent inputs, as well as the neuron response to a sequence of stimuli (asynchronous excitation). Figure 1 illustrates the response of neuron to two (a), three (b) and four (c) non-simultaneous inputs. We provided subthreshold stimuli one by one to the neuron and analyzed the response (i.e. two synapses were simulated with di erent delays). In the case of two asynchronous stimuli the neuron red under either of two conditions: (1) when the period of time between stimuli was less than a certain xed value, dependent on the strength of the synaptic connections or (2) when the sum of synaptic weights was greater than the value of excitation threshold. The neuronal response to a sequence of subthreshold stimuli (more than two) was also examined. In this case we observed the e ect of a gradually depolarizing membrane potential (even in the case of small synaptic weight) and showed that a neuron generated an AP after a sucient number of stimuli was provided. The neuronal response to inhibitory stimuli was also examined. We con rmed that the hyperpolarization of the neuronal membrane potential seen after applying an inhibitory stimulus was not simply the reverse of depolarization. We observed a distinct lesser in uence of inhibitory synapses on membrane potential. With two synchronous stimuli of opposite polarity there was a nonlinear summation at the neurons output. A single excitatory stimulus produced stronger membrane potential change. To prevent generation of the AP at least two simultaneous inhibitory inputs of same strength were needed (Fig. 2). To balance the e ect of EPSPs on membrane potential even more inhibition of neuronal input was required. To prevent any change in resting membrane potential after applying a superthreshold excitatory stimulus (weight value w = 0:34), ten inhibitory stimuli of same strength (each with weight value w = ?0:34) had to be simultaneously applied. 6
Fig. 1. The temporal and spatial summation property of the neuron model. The four top traces represent voltage inputs to the neuron. The bottom trace shows voltage at the neuron output. Two (a), three (b), and four (c) excitatory potentials were activated non-simultaneously. The summation of inputs increases the neuron membrane potential depolarization. The action potential (AP) is not generated until the neuron membrane potential reaches the threshold level (excitation is strong enough). Note: the AP (c) was truncated to t on this gure. Parameters : w = 0:10, o = 2:5 ms, d = 3:5 ms, EEPSPs = 55 mV; for values of other parameters see appendix.
3.2 Neuronal output properties for train input stimuli
The response of the neuron to a single superthreshold stimulus was also tested. We con rmed that the model output for train stimuli follow systematic rules. These output properties are described by the interspike-interval curve (Fig.3). The neural response depended both on the interval between successive stimuli and the duration of the stimulus. We examined the response time of a neuron in relation to the time interval between a pair of stimuli. Figure 3 shows how the timing of a response to the second stimulus (t - vertical axis) depends on the interval between the two stimuli (t - horizontal axis). Plots were obtained for various values of t with two di erent synaptic weights w. The refractory phase, when the neuron did not re an AP ( t ! 1 ), is clearly visible for small t on both panels of Fig. 3. Moreover, in a certain range of parameter w - synaptic strength, a second region, when the neuron was refractory, was 2
1
1
2
1
7
Fig. 2. Illustration of simultaneously occurring excitation and inhibition at the neuron input. The two top traces represent voltage inputs to the neuron; the shows the excitatory input, the second trace shows the sum of inhibitory inputs. The bottom trace shows voltage at the neuron output. One (a), two (b), and four (c) inhibitory potentials were activated simultaneously with one excitatory potential. At least two simultaneous inhibitory inputs are needed to cancel out one excitatory input of same strength and prevent generation of the AP. The synaptic weight parameter w = 0:45 for excitatory synapse and w = ?0:45 for inhibitory synapse, EEPSPs = 55 mV, EIPSPs = ?72, o = 2:5 ms, d = 3:5 ms; for values of other parameters see the Appendix.
identi ed. Both plots have distinct minima of t , which represent the shortest response time following a stimulus. These model properties are in accordance with observed physiological phenomena of spatial and temporal summation in various experimental conditions (Abeles 1991; Awiszus 1989; Nicholls et al. 1992; Kandel et al. 1993). 2
8
Fig. 3. Interspike-interval curve illustrates neuronal output behavior for train input stimuli. Two superthreshold stimuli were applied at the model neuron input. The interval between stimuli - t1 is shown on the horizontal axis and the response time after second stimulus - t2 is shown on the vertical axis . The value of synaptic weight was set at twice that of the threshold value. An interval of 1 ms corresponds to 100 steps of computation. Diamonds show results for a synaptic weight w = 0:8; there is a short refractory period immediately after stimulus. Triangles show plot for value of synaptic weight w = 0:34; there is the second refractory period (between 7 and 10 ms), when the neuron will not produce APs after the stimulus. Synaptic Paremeters: EEPSPs = 55 mV, o = 2:5 ms, d = 3:5 ms; other parameters as in the Appendix.
3.3 Two neuron excitatory loop
To analyze the characteristics of bursting activity in the absence of inhibition, we simulated a two-neuron excitatory loop with background synaptic input. The output of one neuron was connected to the input of the second neuron and vice versa. We initially set one neuron to the active state and investigated the dependence of AP generation on the strength of excitatory synaptic weight w. We found that the system would generate APs inde nitely when the value of w was larger than 0.34 (with other synaptic parameters value as in Fig. 4) for both synapses. For smaller values of w the generation of APs is not sustained after some time. In the case of two neurons initially in the active state, the system is bi-stable and its state depends on the connection weight and synaptic delay value. Neurons active at t = 0 may reach an inactive state, 9
or may continuously generate APs. This depends on the value of the synaptic weight of synaptic connections between neurons. In both cases, adding the background activity to input may cause the system to switch from one state to another (Fig. 4). We observed a simple mechanism of cessation of continuous activity that resulted from a neuronal output properties for train input stimuli. The high activity intervals may start and stop after the background input spike is red. To show that the random external input may start the periodic activity, we set one neuron to the active state at t = 0, with synaptic weights w > 0:34 and modeled background synaptic inputs with = 0:00007. Figure 4 illustrates that this degree of external excitatory synaptic input could trigger synchronous recurrent bursting in these two connected neurons. The
Fig. 4. An example of the simulated burst activity of a two neuron loop in a voltage representation. Arrows indicate an external random AP arriving at an input to neuron A. With the criteria described in text, B illustrates the activity in neuron B which is synchronous with neuron A. An epoch of a 450 ms simulation is shown. Some of the random background APs initiate bursts in both neurons, while others cause the end of bursts. Synaptic parameters: w = 0:72, o = 2:5 ms, d = 3:5 ms; Background activity parameter = 0:00007; other parameters are as in Appendix.
background activity also in uences the duration of the bursting activity. The duration of the intervals between burst activity is increased by decreasing the probability . If the probability is too low(< 0:00001), a bursting activity is not likely to stop after being triggered and the system generates APs inde nitely. If is too high(> 0:01), bursting activity is also not likely to stop because new activity is being continuously triggered. 10
4 Discussion The model proposed here is intentionally a simple one, incorporating certain important membrane properties, but with simple nonquantal synaptic inputs onto the cell soma. The extension of the neuron model by adding a simple synaptic current in the form of Eq. 8, allows us to demonstrate di erent patterns of neuron activity. The proposed synaptic connection model has no in uence on the basic features of the original neuron model. The excitability and oscillatory behaviors in the presence of superthreshold excitation are the same as in the original Av Ron-Rinzel model. Moreover, by introducing synaptic current, di erent features of the model can be studied, such as temporal and spatial integration of inputs, and neuronal output properties for train input stimuli. The neuron always res when the sum of the synaptic inputs is greater than the value of the threshold excitation. This suggested that neuron response for multiexcitatory stimulus, in this model, has a linear dynamic within this range of parameters. Simultaneously occurring excitation and inhibition of the neuron revealed that the relationship between excitatory and inhibitory synapses is not a simple linear one. This can be explained by analysis of the synaptic model properties. Synaptic current is generated by a time-dependent conductance increase (Eq. 9) in association with a reversal potential Esyn (Eq. 8). To simulate the action of synapses correctly, it is necessary to incorporate both the membrane potential and conductance changes. Employing conductance changes to generate synaptic current reproduces two important properties of synaptic interaction, the membrane potential change of PSPs and a change in input conductance (both of which play important roles in summation). The potential change produced by a PSP is important to bring the membrane potential either closer to threshold or farther from threshold, but the magnitude of the potential change does not necessarily re ect the strength of the connection. For example, an inhibitory synapse with a reversal potential close to a resting membrane potential may produce only small IPSPs, but the resulting conductance change can be large enough to cause powerful inhibition. The magnitude of the conductance change in uences not only the size of PSPs generated for a given input but also the magnitude of other concurrent PSPs from other inputs, thus leading to nonlinear spatial summation depending upon the value of parameters used in the synaptic model (Eqs. 8 and 9). The response of the neuron to a sequence of subthreshold stimuli (asynchronous excitation) leads to the ring of the neuron after a sucient number of stimuli are provided. These simulations show that our model neuron has properties of spatial and temporal summation similar to those of a real neuron. Interesting results are also seen during stimulation of a neuron through a 11
synapse with small synaptic weight w = 0:34. In a certain range of the synaptic weight values, we found a second region (di erent then the rst refractory phase) where the neuron would not re an AP (Fig. 3). This feature was determined to be important in the mechanism of burst cessation in the absence of inhibition. In a simple excitatory loop of two neurons, without inhibitory connections, we observed slow changes in the activity of neurons that resulted from the in uence of the background activity and neuronal interaction. Periods of high activity emerge alternately with silent epochs and the characteristic time constants of these oscillations are of a magnitude larger than responses of a single neuron. In case of extremely low levels of background activity ( < 0:00005), we noticed that periods of high and low activity can last a several hundred milliseconds. This nding suggests the possible role of network interaction in generating slow wave activity. Slow changes of activity can be observed in a wide range of biological neural networks, from dissociated spinal cord cell cultures (Franaszczuk and Bergey 1991) to the intact cerebral cortex (e.g. -waves, Speckman and Elger 1993). The duration of such slow oscillations cannot be easily derived from classical IPSPs or EPSPs (which only last less then 100 ms). Whether slow waves basically emerge from interactions in a network of neurons, or are a direct consequence of intrinsic membrane kinetics of single neurons is controversial (Lopes da Silva 1993). Activation of ion channels with slow calcium-mediated potassium conductances are a one candidate for producing this phenomena (Traub and Wong 1982). Our ndings show that simple neural units without these long-lasting membrane processes are still capable of generating long lasting activity. The ndings from these studies support the hypothesis that the prolonged transients of activity, observed in neurons, can be a consequence of neuronal interactions, as well as of certain intrinsic membrane properties of single neurons (e.g. slow calcium-mediated potassium current changes). Synchronized bursting in neuronal networks can result from recurrent excitatory synapses. The simple two neuron model here is an example of burst generation dependent upon such a recurrent excitatory loop. In addition our simulation illustrates one mechanism of burst generation dependent upon background (external) synaptic activity. This random activity is a primary cause of switching between discrete bursts and the quiescent interburst interval in a loop of two neurons. The results of a bursting activity simulation in larger networks will be the subject of a forthcoming raport. The results presented in this paper for a two neuron loop, suggest that an increase in background activity (represented by ) tends to cause decreased activity of the network. While various oscillatory behaviors may underlie a variety of normal cerebral 12
functions, the abrupt transition to synchronized bursting of neuronal networks is a characteristic of the pathologic state of epileptic seizure generation. Understanding the mechanisms that lead to onset of such synchronous excitation can provide important insights, and has been addressed with various model systems (Traub and Dingledine, 1990, Traub and Miles, 1991). Much less studied, but perhaps of equal importance, are the underlying mechanisms that spontaneously terminate the synchronous bursting of seizure activity. This simple two neuron model here provides two mechanisms for cessation of bursting, independent of synaptic inhibition (network or background). Since synchronized bursting may result from situations where there is failure of inhibitory mechanisms and resultant increased excitation, it is important to identify potential mechanisms of burst cessation that do not depend upon inhibitory activity. Our simulations of a two neuron loop shows that there are three situations when synchronous bursting stops. The rst occurs when a background spike arrives at the rst neuron, while the second neuron is in the refractory phase. This causes the cessation of cyclic AP generation and establishes an inactive state (as shown in Fig. 4). This concurs with the ndings of van Ooyen and al.(1992) and Segal and Furshpan (1990). In addition, for a certain range of parameter w (synaptic weight) there are other periods when the neurons will not re after a stimulus (see Fig. 3). In this instance a background spike does not directly cause the burst cessation, but instead disturbs the cyclic AP generation. After ring spontaneously, the rst neuron goes through the sequence of two refractory states. If input from the second neuron occurs while the rst neuron cannot produce APs (second refractory phase, visible in Fig. 3), the second neuron receives no input, and the cyclic generation of APs terminates. This accounts for burst cessation in case of a weak or moderate value of synaptic weight in the neuronal loop. Figure 5a shows the phase plane representation of one of the neurons at the end of the burst and illustrates the third possibility. The system does not achieve the usual steady state, but instead is forced to oscillate in a high W state (refractory state), which leads to a degenerated low voltage oscillation and a return to the steady state point (Fig. 5b). This mechanism accounts for interburst intervals of moderate duration (up to 20 ms) and happens when a large synaptic excitation in a neuronal loop occurs (the value of parameters w synaptic weight was set of a magnitude larger than those of a threshold value). The length of such an interval is a multiple of the period of degenerated oscillation. The simple two neuron model studied here with background synaptic input, proved adequate to simulate a variety of properties of excitatory neurons that contribute to the production of synchronous bursting. In addition this system 13
Fig. 5. a. The phase plane representation of a modeled neuron in the case of a large value for synaptic weight. The trajectories in upper part of plot represent the refractory phase of the neuron (high values of W ). The neuron does not achieve the resting potential. The oscillation occurs for a high value of the membrane potential V . The time course of potential V for both neurons. The over-excitation leads to an end of cyclic AP generation during burst activity. Parameters: w = 9:0, synaptic delay = 0.75 ms, others as in Appendix.
14
provides illustrations of cessation of such bursting produced by the external background excitatory synaptic input. We believe that these methods can be useful in modeling larger and more complex networks with good computational eciency.
Appendix Basic set of parameters used in model computations
Cm gNa gK gL VNa VK VL aW V =W am V =m
mp wp s (
)
( ) 1 2
(
)
( ) 1 2
= 1 F/cm = 120 mS/cm = 36 mS/cm = 0.3 mS/cm = 55 mV = -72 mV = -49.4 mV = 0.045 = -55 mV = 0.055 = -33 mV = 0.2 =3 =4 = 1.3 2
2
2
2
Synaptic parameters
synaptic delay w o d EEPSPs EIPSPs
0 ms < delay < 4.0 ms 0.1 < w < 1.0 0.5 ms < o < 3 ms 3 ms < d < 5 ms 10 mV < EEPSPs < 55 mV = -72 mV 15
Background activity parameter
0:00005 < < 0:001
Acknowledgement The authors wish to thank Dr. S. Judge for her helpful comments and suggestions. These investigations were supported in part by NIH grant NS 33732-01 to G. K. Bergey and P. J. Franaszczuk. P. Kudela was supported by KBN grant 3P401-00407.
References Abeles M (1991) Corticonics: neural circuits of the cerebral cortex. Cambridge Univ. Press, Cambridge, pp228-229. Av-Ron E, Parnas H, Segel LA (1991) A minimal biophysical model for an excitable and oscillatory neuron. Biol Cybern 65:487-500. Awiszus F (1989) On the description of neuronal output properties using spike train data. Biol Cybern 60:323-333. Chay TR, Lee YS (1990) Bursting, beating and chaos by two functionally distinct inward current inactivation in excitable cell. Ann.of the New York Acad.of Sci., Vol.591,328-350 FitzHugh R (1961) Impulses and physiological states in model of nerve membrane. Biophys J 1:445-466. Franaszczuk PJ, Bergey GK(1991). Patterns of spontaneous postsynaptic potentials during organized convulsant activity produced by tetanus toxin in cultured mouse spinal cord neurons. Epilepsia 32 (Suppl 3): 66. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol (London) 117:500-544. Hop eld JJ, Tank DW (1986). Computing with neural circuits: a model. Science 233:625-33. Kandel ER, Schwartz JH, Jessel TM. (1993) Principles of neural science. (3rd ed.) Elsevier, New York, Amsterdam, London, Tokyo.pp 34-119. Lopes da Silva FH (1993) Dynamics of EEGs as signals of neuronal populations: Models and theoretical considerations. In: Electroencephalography. Ba16
sic principles, clinical applications, and related elds.(3rd ed.) E. Niedermeyer and FH Lopes da Silva (Eds.)Wiliams & Wilkins, Baltimore,MD. pp63-77. Nicholls JG, Martin AR, Wallace BG (1992) From neurons to brain: a cellular and molecular approach to the function of the nervous system. Sinauer Associates, Inc. 3rd edn. Dinauer Associates, Sunderland, Mass. Ooyen van A, Pelt van J, Corner MA, Lopes da Silva FH (1992) The emergence of long-lasting transients of activity in simple neural networks. Biol Cybern 67:269-277. Rinzel J (1985) Excitation dynamics: insight from simpli ed membrane models. Fed. Proc. 44:2844-2946. Rinzel J, Ermentrout GB (1989) Analysis of neural excitability and oscillations. In: Koch C, Segev I (eds) Methods in neuronal modeling. MIT Press, Massachusetts, pp 135-169. Segal MM, Furshpan EJ (1990) Epileptiform activity in microcultures containing small numbers of hippocampal neurons. J Neurophysiol 64:1390-1399. Speckmann EJ, Elger CE (1993) Introduction to the Neurophysiological basis of the EEG and DC potentials. In: Electroencephalography. Basic principles, clinical applications, and related elds.(3rd edition) E. Niedermeyer and FH Lopes da Silva (Eds.)Wiliams & Wilkins, Baltimore,MD. pp15-26.. Traub RD, Wong RKS (1982) Cellular mechanism of neuronal synchronization in epilepsy. Science 216:745-747. Traub RD, Miles R (1991) Neuronal networks of the hippocampus. Cambridge Univ. Press, Cambridge, UK. Traub RD, Dingledine R (1990) Model of synchronized epileptiform bursts induced by high potassium in the CA3 region of the rat hippocampal slice: role of spontaneous EPSPs in initiation. J. Neurophysiol. 64:1009-1018. Yamada WM, Koch C, Adams PR (1989) Multiple channels and calcium dynamics. In: Koch C., Segev I. (eds) Methods in neuronal modeling. MIT Press, Massachusetts, pp 97-133.
17
a;1
Piotr J. Franaszczuk
Gregory K. Bergey
b;2
b;c;2
Laboratory of Medical Physics, Institute of Experimental Physics, Warsaw University, Warsaw (Poland) b Maryland Epilepsy Center, Department of Neurology, University of Maryland School of Medicine and Medical Center, 22 South Greene Street,Baltimore, MD 21201 (USA) c Department of Physiology, University of Maryland School of Medicine, Baltimore, MD 21201 (USA) a
Abstract The space-lumped two variable neuron model is studied. The extension of the neural model by adding a simple synaptic current allows to demonstrate neural interactions. The production of synchronous burst activity in this simple two neuron excitatory loop is modeled, including the in uence of random background excitatory input. The ability of the neuron model to spatially and temporally integrate inputs is shown. Two refractory periods after stimuli were identi ed and their role in burst cessation is shown. Our ndings show that simple neural units without long-lasting membrane processes are capable of generating long lasting patterns of activity. The results of simulation of simple background activity suggest that an increase in background activity tends to cause decreased activity of the network. This phenomenon as well as existence of two refractory periods allows for burst cessation without inhibition in this simple model.
1 Introduction When modeling neural networks, we are usually interested in models that are as simple as possible with regard to computational complexity. On the other hand we would like the models to exhibit certain physiological features. The 1 2
Supported by KBN grant 3P401-00407 Supported by NIH grant NS 33732-01
Preprint submitted to Biological Cybernetics
April 1997
simple spin- ip model (Hop eld and Tank 1986) is easy to implement, but has rather limited physiological value. The choice of model is often a compromise between computing ability and the complexity of the phenomena that the system is intended to simulate. There are many neuronal models derived from the classical Hodgkin-Huxley (1952) model. Some of them, (e.g. Yamada et al. 1989; Chay and Lee 1990), focus on detailed quantitative modeling of ion channels dynamics. Others (e.g. FitzHugh 1961; Rinzel 1985; Av-Ron et al. 1991) concentrate on general, qualitative neuronal properties, such as the generation of individual or repetitive action potentials. Models of neurons from the rst group are more complex and were introduced to model the electrical activity of the individual type of cells. Neuron models belonging to the second group are usually much easier to implement and are simpler to analyze, but are missing some of the physiological features. Our goal is to implement a relatively simple neuron model, with certain physiological membrane features, and to investigate how the modeled neuron behaves when connected to other similar neurons. We will show that, despite their simplicity, such connected neurons exhibit various phenomena observed in real neural networks. It should be noted that this kind of reduced neuron model (Rinzel 1985; Av-Ron et al. 1991) has been studied in the context of the single neuron. This approach limits analysis of the characteristics of the model and might hide some neuronal properties. For example, Rinzel (1985) discusses the phenomena of threshold and repetitive ring in response to a single brief current pulse. In real systems, synaptically connected neurons excite or inhibit each other. Our goal is to study the behavior of a modeled neuron in conditions similar to physiological reality. As a starting point we took the Rinzel's (1985) space-lumped two-variable model. Next we modeled the synaptic input in this model. We examined the ability of the neuron to spatially and temporally integrate inputs. We also analyzed the neuronal output properties for various input stimuli. Then, we added the simulated random background activity to input, and connected two neurons into the feedback loop. Finally, we examined di erent patterns of neural activity, especially burst onset and cessation in the simple neuron loop.
2 Model description 2.1 Model of neuron
We used Rinzel's model of a single neuron as modi ed by Av-Ron et al. (1991). This choice is justi ed by the easier analytical tractability of this model, and its ability to generate di erent types of potentials. The model is based on a simpli cation of the Hodgkin-Huxley model (1952) of excitable membrane (H2
H). Here, following the observation of FitzHugh (1961) that the sum of two H-H variables n and h during an action potential is constant, one common variable is introduced. The second simpli cation in this model is an assumption that the third "fast" Hodgkin-Huxley m variable is in a quasi-steady state m with respect to changing the value of the membrane potential. This approach simpli es the model by reducing the number of variables, while not signi cantly altering the model dynamics. The Av Ron's model is composed of two ordinary di erential equations. The rst equation describes the rate of change of voltage with respect to time as proportional to the sum of three currents owing through the membrane: an inward sodium current INa, an outward potassium current IK , and a leak current IL. Each ion current Ii , is described by a maximal conductance gi and a driving force proportional to (V ? Vi). Formulas de ning currents INa and IK include activation-deactivation terms. The second equation describes the rate of change with respect to time of the recovery component W , a linear combination of two Hodgkin-Huxley variables n and h. W is called the recovery variable because it is responsible for the conclusion of the action potential and return to resting membrane potential. A brief description of the model follows while a more detailed discussion of this neuron model can be found in Av Ron et al. (1991). Denoting membrane potential as V and Rinzel's "recovery" variable as W the basic set of equations is as follows:
Cm dV=dt = I ? gNa mmp 1 (V )(1 ? W )(V ? VNa ) ? gK (W=s)wp(V ? VK ) ? gL(V ? VL)
(1) (2)
dW=dt = (W1(V ) ? W )= (V )
(3)
and
where the parameters mp = 3, wp = 4, and s = 1:3 and W1(V ) and m1(V ) are steady state conductance functions de ned by:
W1(V ) = (1 + exp [?2a W (V ? V =W ])?
(4)
m1(V ) = (1 + exp [?2a m (V ? V =m ])?
(5)
(
(
)
)
( ) 1 2
( ) 1 2
1
1
and
(V ) = ( exp [a W (V ? V =W )] + exp [?a W (V ? V =W )])? (
( ) 1 2
)
(
)
( ) 1 2
1
3
(6) (7)
where is the relaxation time function for recovery variable W . Terms 4 and 5 have sigmoid curve shapes, with the half maximal voltage value of V = and the slope of the curve controlled by a. All the parameters a W , a m , V =W , V =m , and g are chosen to t the experimental data (see Appendix). This model exhibits two types of qualitative behavior. For a short suprathreshold stimulus the model will generate a single action potential. The current I in Eq.1 represents external input stimuli. In our simulations this accounts for input from other neurons. There are other input and parameter values, for which the model yields oscillatory behaviors (e.g. limit cycle oscillations, AvRon et al. 1991). 1 2
(
)
(
)
( ) 1 2
( ) 1 2
Each neuron has a probability of generating a postsynaptic potential at each simulated time interval, independent of synaptic inputs. This simulates spikes arriving from other neurons not modeled, and will be referred to as background activity. The above de nition implies that generation of background activity is a random Poisson process. 2.2 Model of the synaptic connection
Our modeling of neuron connections is based upon the assumption that the neuron integrates inputs from other neurons that generate postsynaptic potentials (PSPs). Inputs can have di erent strengths modeled by synaptic weight and di erent polarity: positive for excitatory inputs, or negative for inhibitory inputs. All delays in transmission of an AP between ring in one neuron and generation of a PSPs in the next neuron are modeled by a single model parameter, the synaptic delay. The synaptic contribution by an external current Isyn is de ned by:
X wj gj(t) (V ? Esyn)
(8)
XN (exp((t ? t)= d) ? exp((ti ? t)= o))
(9)
Isyn = and
g(t) =
Nsyn j =1
i=1
where d and o are respectively, decay and onset parameters, i denotes the time of arrival of i-th (after transmission and synaptic delay); Nsyn is the number of input synapses; and wj stands for the synaptic weight, positive for excitatory and negative for inhibitory connections; Esyn is the synaptic reversal potential. This form of synaptic conductance allows for adjustment of the time course of evoked PSPs by changing values of d and o . Summation 4
over time (Eq.9) is limited to N most recent APs, for which t ? ti is within a preset interval. This interval depends on the decay parameter d . For a d ranging between 2 and 3 milliseconds we limit the summation interval to a range of 15 - 20 milliseconds. For each red AP a synaptic current is added to the postsynaptic neuronal equation after preset synaptic delay. Each neuron can have multiple input synapses and di erent neurons can have di erent number of synapses. Output from each neuron can be sent to several other neurons. Each synaptic connection can have a di erent weight and delay. In this way multiple neuron con gurations can be simulated. In this paper we describe only the simplest interactions between two neurons. The dynamics of simulated neurons is illustrated by the voltage potential representation (plot shows a value V for each modeled neuron in time).
2.3 Computer algorithms and codes
The model equation, (Eq.1 and 3) is integrated using an explicit rst-order Euler scheme to update the neuron membrane voltage, according to the following formula:
V (t + t) = V (t) + (dV (t)=dt) t
(10)
Integration step t is set to 10 s. We choose to use this simple scheme to speed up computation with minimal loss of precision. This is checked by using Runge-Kutta method for integration, and the results of simulation are not signi cantly changed. All simulation programs were written in ANSI C, compiled and run on both the IBM RS6000 workstation running AIX operating system and the PC DOS computers. The main program accepts input parameters from a disk le and writes output data to a disk le. The input data de nes both the neuron model parameters and interneuronal connections. For this simulation, separate programs for generating input parameters and for displaying output data were developed. This allowed the main simulation to be computed on a powerful UNIX workstation, while design of the network and analysis of output could be done interactively on PC computers. Supporting procedures were developed to create input les for the simulation module and to display di erent kinds of output data. 5
3 Results of simulations
3.1 Temporal and spatial summation property of the model
First we examined the ability of our implementation of the neuron model to reproduce basic physiological phenomena. The model can provide an AP in response to a superthreshold stimulus. For a standard input stimulus above in which an AP appears, we determined the threshold value for synaptic weight. We then examined the summation properties of our model with two or more synaptic excitatory stimuli, from di erent inputs, as well as the neuron response to a sequence of stimuli (asynchronous excitation). Figure 1 illustrates the response of neuron to two (a), three (b) and four (c) non-simultaneous inputs. We provided subthreshold stimuli one by one to the neuron and analyzed the response (i.e. two synapses were simulated with di erent delays). In the case of two asynchronous stimuli the neuron red under either of two conditions: (1) when the period of time between stimuli was less than a certain xed value, dependent on the strength of the synaptic connections or (2) when the sum of synaptic weights was greater than the value of excitation threshold. The neuronal response to a sequence of subthreshold stimuli (more than two) was also examined. In this case we observed the e ect of a gradually depolarizing membrane potential (even in the case of small synaptic weight) and showed that a neuron generated an AP after a sucient number of stimuli was provided. The neuronal response to inhibitory stimuli was also examined. We con rmed that the hyperpolarization of the neuronal membrane potential seen after applying an inhibitory stimulus was not simply the reverse of depolarization. We observed a distinct lesser in uence of inhibitory synapses on membrane potential. With two synchronous stimuli of opposite polarity there was a nonlinear summation at the neurons output. A single excitatory stimulus produced stronger membrane potential change. To prevent generation of the AP at least two simultaneous inhibitory inputs of same strength were needed (Fig. 2). To balance the e ect of EPSPs on membrane potential even more inhibition of neuronal input was required. To prevent any change in resting membrane potential after applying a superthreshold excitatory stimulus (weight value w = 0:34), ten inhibitory stimuli of same strength (each with weight value w = ?0:34) had to be simultaneously applied. 6
Fig. 1. The temporal and spatial summation property of the neuron model. The four top traces represent voltage inputs to the neuron. The bottom trace shows voltage at the neuron output. Two (a), three (b), and four (c) excitatory potentials were activated non-simultaneously. The summation of inputs increases the neuron membrane potential depolarization. The action potential (AP) is not generated until the neuron membrane potential reaches the threshold level (excitation is strong enough). Note: the AP (c) was truncated to t on this gure. Parameters : w = 0:10, o = 2:5 ms, d = 3:5 ms, EEPSPs = 55 mV; for values of other parameters see appendix.
3.2 Neuronal output properties for train input stimuli
The response of the neuron to a single superthreshold stimulus was also tested. We con rmed that the model output for train stimuli follow systematic rules. These output properties are described by the interspike-interval curve (Fig.3). The neural response depended both on the interval between successive stimuli and the duration of the stimulus. We examined the response time of a neuron in relation to the time interval between a pair of stimuli. Figure 3 shows how the timing of a response to the second stimulus (t - vertical axis) depends on the interval between the two stimuli (t - horizontal axis). Plots were obtained for various values of t with two di erent synaptic weights w. The refractory phase, when the neuron did not re an AP ( t ! 1 ), is clearly visible for small t on both panels of Fig. 3. Moreover, in a certain range of parameter w - synaptic strength, a second region, when the neuron was refractory, was 2
1
1
2
1
7
Fig. 2. Illustration of simultaneously occurring excitation and inhibition at the neuron input. The two top traces represent voltage inputs to the neuron; the shows the excitatory input, the second trace shows the sum of inhibitory inputs. The bottom trace shows voltage at the neuron output. One (a), two (b), and four (c) inhibitory potentials were activated simultaneously with one excitatory potential. At least two simultaneous inhibitory inputs are needed to cancel out one excitatory input of same strength and prevent generation of the AP. The synaptic weight parameter w = 0:45 for excitatory synapse and w = ?0:45 for inhibitory synapse, EEPSPs = 55 mV, EIPSPs = ?72, o = 2:5 ms, d = 3:5 ms; for values of other parameters see the Appendix.
identi ed. Both plots have distinct minima of t , which represent the shortest response time following a stimulus. These model properties are in accordance with observed physiological phenomena of spatial and temporal summation in various experimental conditions (Abeles 1991; Awiszus 1989; Nicholls et al. 1992; Kandel et al. 1993). 2
8
Fig. 3. Interspike-interval curve illustrates neuronal output behavior for train input stimuli. Two superthreshold stimuli were applied at the model neuron input. The interval between stimuli - t1 is shown on the horizontal axis and the response time after second stimulus - t2 is shown on the vertical axis . The value of synaptic weight was set at twice that of the threshold value. An interval of 1 ms corresponds to 100 steps of computation. Diamonds show results for a synaptic weight w = 0:8; there is a short refractory period immediately after stimulus. Triangles show plot for value of synaptic weight w = 0:34; there is the second refractory period (between 7 and 10 ms), when the neuron will not produce APs after the stimulus. Synaptic Paremeters: EEPSPs = 55 mV, o = 2:5 ms, d = 3:5 ms; other parameters as in the Appendix.
3.3 Two neuron excitatory loop
To analyze the characteristics of bursting activity in the absence of inhibition, we simulated a two-neuron excitatory loop with background synaptic input. The output of one neuron was connected to the input of the second neuron and vice versa. We initially set one neuron to the active state and investigated the dependence of AP generation on the strength of excitatory synaptic weight w. We found that the system would generate APs inde nitely when the value of w was larger than 0.34 (with other synaptic parameters value as in Fig. 4) for both synapses. For smaller values of w the generation of APs is not sustained after some time. In the case of two neurons initially in the active state, the system is bi-stable and its state depends on the connection weight and synaptic delay value. Neurons active at t = 0 may reach an inactive state, 9
or may continuously generate APs. This depends on the value of the synaptic weight of synaptic connections between neurons. In both cases, adding the background activity to input may cause the system to switch from one state to another (Fig. 4). We observed a simple mechanism of cessation of continuous activity that resulted from a neuronal output properties for train input stimuli. The high activity intervals may start and stop after the background input spike is red. To show that the random external input may start the periodic activity, we set one neuron to the active state at t = 0, with synaptic weights w > 0:34 and modeled background synaptic inputs with = 0:00007. Figure 4 illustrates that this degree of external excitatory synaptic input could trigger synchronous recurrent bursting in these two connected neurons. The
Fig. 4. An example of the simulated burst activity of a two neuron loop in a voltage representation. Arrows indicate an external random AP arriving at an input to neuron A. With the criteria described in text, B illustrates the activity in neuron B which is synchronous with neuron A. An epoch of a 450 ms simulation is shown. Some of the random background APs initiate bursts in both neurons, while others cause the end of bursts. Synaptic parameters: w = 0:72, o = 2:5 ms, d = 3:5 ms; Background activity parameter = 0:00007; other parameters are as in Appendix.
background activity also in uences the duration of the bursting activity. The duration of the intervals between burst activity is increased by decreasing the probability . If the probability is too low(< 0:00001), a bursting activity is not likely to stop after being triggered and the system generates APs inde nitely. If is too high(> 0:01), bursting activity is also not likely to stop because new activity is being continuously triggered. 10
4 Discussion The model proposed here is intentionally a simple one, incorporating certain important membrane properties, but with simple nonquantal synaptic inputs onto the cell soma. The extension of the neuron model by adding a simple synaptic current in the form of Eq. 8, allows us to demonstrate di erent patterns of neuron activity. The proposed synaptic connection model has no in uence on the basic features of the original neuron model. The excitability and oscillatory behaviors in the presence of superthreshold excitation are the same as in the original Av Ron-Rinzel model. Moreover, by introducing synaptic current, di erent features of the model can be studied, such as temporal and spatial integration of inputs, and neuronal output properties for train input stimuli. The neuron always res when the sum of the synaptic inputs is greater than the value of the threshold excitation. This suggested that neuron response for multiexcitatory stimulus, in this model, has a linear dynamic within this range of parameters. Simultaneously occurring excitation and inhibition of the neuron revealed that the relationship between excitatory and inhibitory synapses is not a simple linear one. This can be explained by analysis of the synaptic model properties. Synaptic current is generated by a time-dependent conductance increase (Eq. 9) in association with a reversal potential Esyn (Eq. 8). To simulate the action of synapses correctly, it is necessary to incorporate both the membrane potential and conductance changes. Employing conductance changes to generate synaptic current reproduces two important properties of synaptic interaction, the membrane potential change of PSPs and a change in input conductance (both of which play important roles in summation). The potential change produced by a PSP is important to bring the membrane potential either closer to threshold or farther from threshold, but the magnitude of the potential change does not necessarily re ect the strength of the connection. For example, an inhibitory synapse with a reversal potential close to a resting membrane potential may produce only small IPSPs, but the resulting conductance change can be large enough to cause powerful inhibition. The magnitude of the conductance change in uences not only the size of PSPs generated for a given input but also the magnitude of other concurrent PSPs from other inputs, thus leading to nonlinear spatial summation depending upon the value of parameters used in the synaptic model (Eqs. 8 and 9). The response of the neuron to a sequence of subthreshold stimuli (asynchronous excitation) leads to the ring of the neuron after a sucient number of stimuli are provided. These simulations show that our model neuron has properties of spatial and temporal summation similar to those of a real neuron. Interesting results are also seen during stimulation of a neuron through a 11
synapse with small synaptic weight w = 0:34. In a certain range of the synaptic weight values, we found a second region (di erent then the rst refractory phase) where the neuron would not re an AP (Fig. 3). This feature was determined to be important in the mechanism of burst cessation in the absence of inhibition. In a simple excitatory loop of two neurons, without inhibitory connections, we observed slow changes in the activity of neurons that resulted from the in uence of the background activity and neuronal interaction. Periods of high activity emerge alternately with silent epochs and the characteristic time constants of these oscillations are of a magnitude larger than responses of a single neuron. In case of extremely low levels of background activity ( < 0:00005), we noticed that periods of high and low activity can last a several hundred milliseconds. This nding suggests the possible role of network interaction in generating slow wave activity. Slow changes of activity can be observed in a wide range of biological neural networks, from dissociated spinal cord cell cultures (Franaszczuk and Bergey 1991) to the intact cerebral cortex (e.g. -waves, Speckman and Elger 1993). The duration of such slow oscillations cannot be easily derived from classical IPSPs or EPSPs (which only last less then 100 ms). Whether slow waves basically emerge from interactions in a network of neurons, or are a direct consequence of intrinsic membrane kinetics of single neurons is controversial (Lopes da Silva 1993). Activation of ion channels with slow calcium-mediated potassium conductances are a one candidate for producing this phenomena (Traub and Wong 1982). Our ndings show that simple neural units without these long-lasting membrane processes are still capable of generating long lasting activity. The ndings from these studies support the hypothesis that the prolonged transients of activity, observed in neurons, can be a consequence of neuronal interactions, as well as of certain intrinsic membrane properties of single neurons (e.g. slow calcium-mediated potassium current changes). Synchronized bursting in neuronal networks can result from recurrent excitatory synapses. The simple two neuron model here is an example of burst generation dependent upon such a recurrent excitatory loop. In addition our simulation illustrates one mechanism of burst generation dependent upon background (external) synaptic activity. This random activity is a primary cause of switching between discrete bursts and the quiescent interburst interval in a loop of two neurons. The results of a bursting activity simulation in larger networks will be the subject of a forthcoming raport. The results presented in this paper for a two neuron loop, suggest that an increase in background activity (represented by ) tends to cause decreased activity of the network. While various oscillatory behaviors may underlie a variety of normal cerebral 12
functions, the abrupt transition to synchronized bursting of neuronal networks is a characteristic of the pathologic state of epileptic seizure generation. Understanding the mechanisms that lead to onset of such synchronous excitation can provide important insights, and has been addressed with various model systems (Traub and Dingledine, 1990, Traub and Miles, 1991). Much less studied, but perhaps of equal importance, are the underlying mechanisms that spontaneously terminate the synchronous bursting of seizure activity. This simple two neuron model here provides two mechanisms for cessation of bursting, independent of synaptic inhibition (network or background). Since synchronized bursting may result from situations where there is failure of inhibitory mechanisms and resultant increased excitation, it is important to identify potential mechanisms of burst cessation that do not depend upon inhibitory activity. Our simulations of a two neuron loop shows that there are three situations when synchronous bursting stops. The rst occurs when a background spike arrives at the rst neuron, while the second neuron is in the refractory phase. This causes the cessation of cyclic AP generation and establishes an inactive state (as shown in Fig. 4). This concurs with the ndings of van Ooyen and al.(1992) and Segal and Furshpan (1990). In addition, for a certain range of parameter w (synaptic weight) there are other periods when the neurons will not re after a stimulus (see Fig. 3). In this instance a background spike does not directly cause the burst cessation, but instead disturbs the cyclic AP generation. After ring spontaneously, the rst neuron goes through the sequence of two refractory states. If input from the second neuron occurs while the rst neuron cannot produce APs (second refractory phase, visible in Fig. 3), the second neuron receives no input, and the cyclic generation of APs terminates. This accounts for burst cessation in case of a weak or moderate value of synaptic weight in the neuronal loop. Figure 5a shows the phase plane representation of one of the neurons at the end of the burst and illustrates the third possibility. The system does not achieve the usual steady state, but instead is forced to oscillate in a high W state (refractory state), which leads to a degenerated low voltage oscillation and a return to the steady state point (Fig. 5b). This mechanism accounts for interburst intervals of moderate duration (up to 20 ms) and happens when a large synaptic excitation in a neuronal loop occurs (the value of parameters w synaptic weight was set of a magnitude larger than those of a threshold value). The length of such an interval is a multiple of the period of degenerated oscillation. The simple two neuron model studied here with background synaptic input, proved adequate to simulate a variety of properties of excitatory neurons that contribute to the production of synchronous bursting. In addition this system 13
Fig. 5. a. The phase plane representation of a modeled neuron in the case of a large value for synaptic weight. The trajectories in upper part of plot represent the refractory phase of the neuron (high values of W ). The neuron does not achieve the resting potential. The oscillation occurs for a high value of the membrane potential V . The time course of potential V for both neurons. The over-excitation leads to an end of cyclic AP generation during burst activity. Parameters: w = 9:0, synaptic delay = 0.75 ms, others as in Appendix.
14
provides illustrations of cessation of such bursting produced by the external background excitatory synaptic input. We believe that these methods can be useful in modeling larger and more complex networks with good computational eciency.
Appendix Basic set of parameters used in model computations
Cm gNa gK gL VNa VK VL aW V =W am V =m
mp wp s (
)
( ) 1 2
(
)
( ) 1 2
= 1 F/cm = 120 mS/cm = 36 mS/cm = 0.3 mS/cm = 55 mV = -72 mV = -49.4 mV = 0.045 = -55 mV = 0.055 = -33 mV = 0.2 =3 =4 = 1.3 2
2
2
2
Synaptic parameters
synaptic delay w o d EEPSPs EIPSPs
0 ms < delay < 4.0 ms 0.1 < w < 1.0 0.5 ms < o < 3 ms 3 ms < d < 5 ms 10 mV < EEPSPs < 55 mV = -72 mV 15
Background activity parameter
0:00005 < < 0:001
Acknowledgement The authors wish to thank Dr. S. Judge for her helpful comments and suggestions. These investigations were supported in part by NIH grant NS 33732-01 to G. K. Bergey and P. J. Franaszczuk. P. Kudela was supported by KBN grant 3P401-00407.
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