The Spike Response Model (SRM) is - just like the nonlinear integrate-and-fire model - a generalization of the leaky integrate-and-fire model. The direction of the generalization is, however, somewhat different. In the nonlinear integrate-and-fire model, parameters are made voltage dependent whereas in the SRM they depend on the time since the last output spike. Another difference between integrate-and-fire models and the SRM concerns the formulation of the equations. While integrate-and-fire models are usually defined in terms of differential equations, the SRM expresses the membrane potential at time t as an integral over the past.
The explicit dependence of the membrane potential upon the last output spike allows us to model refractoriness as a combination of three components, viz., (i) a reduced responsiveness after an output spike; (ii) an increase in threshold after firing; and (iii) a hyperpolarizing spike after-potential. In Section 4.2.1 the Spike Response Model is introduced and its properties illustrated. Its relation to the integrate-and-fire model is the topic of Section 4.2.2. An important special case of the Spike Response Model is the simplified model SRM_{0} that we have already encountered in Chapter 1.3.1. Section 4.2.3 will discuss it in more detail.
In the framework of the Spike Response Model
the state of a neuron i is described by a single variable u_{i}. In the
absence of spikes, the variable u_{i} is at its resting value,
u_{rest} = 0. Each incoming spike will perturb u_{i} and
it takes some time before u_{i} returns to zero. The function
describes the time course of the response to an incoming spike. If, after the
summation of the effects of several incoming spikes, u_{i} reaches the
threshold an output spike is triggered. The form of the action
potential and the after-potential is described by
a function . Let us suppose neuron i has fired its last spike at time
. After firing the evolution of u_{i} is given by
In contrast to the integrate-and-fire neuron discussed in Section (4.1) the threshold is not fixed but may also depend on t -
(t - ) . | (4.25) |
= maxt_{i}^{(f)} < t . | (4.27) |
So far Eqs. (4.1) and (4.24) define a mathematical model. Can we give a biological interpretation of the terms? Let us identify the variable u_{i} with the membrane potential of neuron i. The functions , and are response kernels that describe the effect of spike emission and spike reception on the variable u_{i}. This interpretation has motivated the name `Spike Response Model', SRM for short (Gerstner, 1995; Kistler et al., 1997). Let us discuss the meaning of the response kernels.
The kernel describes the standard form of an action potential of neuron i including the negative overshoot which typically follows a spike (after-potential). Graphically speaking, a contribution is `pasted in' each time the membrane potential reaches the threshold ; cf. Fig. 4.5. Since the form of the spike is always the same, the exact time course of the action potential carries no information. What matters is whether there is the event `spike' or not. The event is fully characterized by the firing time t_{i}^{(f)}. In a simplified model, the form of the action potential may therefore be neglected as long as we keep track of the firing times t_{i}^{(f)}. The kernel describes then simply the `reset' of the membrane potential to a lower value after the spike at just like in the integrate-and-fire model. The leaky integrate-and-fire model is in fact a special case of the SRM as we will see below in Section 4.2.2.
The kernel (t - , s) is the linear response of the membrane potential to an input current. It describes the time course of a deviation of the membrane potential from its resting value that is caused by a short current pulse (``impulse response''). We have already seen in Chapters 2.2 and 3 that the response depends, in general, on the time that has passed since the last output spike at . Immediately after many ion channels are open so that the resistance of the membrane is reduced. The voltage response to an input current pulse decays therefore more rapidly back to zero than in a neuron that has been inactive. A reduced or shorter response is one of the signatures of neuronal refractoriness. This form of refractory effect is taken care of by making the kernel depend, via its first argument, on the time difference t - . We illustrate the idea in Fig. 4.5. The response to a first input pulse at t' is shorter and less pronounced than that to a second one at t'', an effect which is well-known experimentally (Fuortes and Mantegazzini, 1962; Stevens and Zador, 1998; Powers and Binder, 1996).
The kernel (t - , s) as a function of s = t - t_{j}^{(f)} can be interpreted as the time course of a postsynaptic potential evoked by the firing of a presynaptic neuron j at time t_{j}^{(f)}. Depending on the sign of the synapse from j, to i, models either an excitatory or inhibitory postsynaptic potential (EPSP or IPSP). Similarly as for the kernel , the exact shape of the postsynaptic potential depends on the time t - that has passed since the last spike of the postsynaptic neuron i. In particular, if neuron i has been active immediately before the arrival of a presynaptic action potential, the postsynaptic neuron is in a state of refractoriness. In this case, the response to an input spike is smaller than that of an `unprimed' neuron. The first argument of (t - , s) accounts for the dependence upon the last firing time of the postsynaptic neuron.
In order to simplify the notation for later use, it is convenient to introduce the total postsynaptic potential,
Refractoriness may be characterized experimentally by the observation that immediately after a first action potential it is impossible (absolute refractoriness) or more difficult (relative refractoriness) to excite a second spike (Fuortes and Mantegazzini, 1962).
Absolute refractoriness can be incorporated in the SRM by setting the dynamic threshold during a time to an extremely high value that cannot be attained.
Relative refractoriness can be mimicked in various ways; see Fig. 4.5. First, after the spike the membrane potential, and hence , passes through a regime of hyperpolarization (after-potential) where the voltage is below the resting potential. During this phase, more stimulation than usual is needed to drive the membrane potential above threshold. This is equivalent to a transient increase of the firing threshold (see below). Second, and contribute to relative refractoriness because, immediately after an action potential, the response to incoming spikes is shorter and, possibly, of reduced amplitude (Fuortes and Mantegazzini, 1962). Thus more input spikes are needed to evoke the same depolarization of the membrane potential as in an `unprimed' neuron. The first argument of the function (or function) allows us to incorporate this effect.
From a formal point of view, there is no need to interpret the variable u as the membrane potential. It is, for example, often convenient to transform the variable u so as to remove the time-dependence of the threshold. In fact, a general Spike Response Model with arbitrary time-dependent threshold (t - ) = + (t - ), can always be transformed into a Spike Response Model with fixed threshold by a change of variables
In Chapter 3 we have studied the FitzHugh-Nagumo model as an example of a two-dimensional neuron model. Here we want to show that the response of the FitzHugh-Nagumo model to a short input current pulse depends on the time since the last spike. Let us trigger, in a simulation of the model, an action potential at t = 0. This can be done by applying a short, but strong current pulse. The result is a voltage trajectory of large amplitude which we identify with the kernel (t). Figure 4.6 shows the hyperpolarizing spike after-potential which decays slowly back to the resting level. To test the responsiveness of the FitzHugh-Nagumo model during the recovery phase after the action potential, we apply, at a time t^{(f)} > 0, a second short input current pulse of low amplitude. The response to this test pulse is compared with the unperturbed trajectory. The difference between the two trajectories defines the kernel (t - , t - t^{(f)}). In Fig. 4.6 several trajectories are overlayed showing the response to stimulation at t = 10, 15, 20, 30 or 40. The shape and duration of the response curve depends on the time that has passed since the initiation of the action potential. Note that the time constant of the response kernel is always shorter than that of the hyperpolarizing spike after-potential. Analogous results for the Hodgkin-Huxley model will be discussed below in Section 4.3.1.
Motoneurons exhibit a rather slow return to the resting potential after an
action potential (Powers and Binder, 1996). The time constant of the
decay of the hyperpolarizing spike after-potential can be in the range of
100ms or more and is therefore much slower than the membrane time constant
that characterizes the response to a short current input. On the other hand,
it is found that if motoneurons are stimulated by a constant super-threshold
current, their membrane potential has a roughly linear trajectory when
approaching threshold. To qualitatively describe these observations, we can
use a Spike Response Model with the following kernels:
The effect of the modulation of the input conductance as a function of t - is depicted in Fig. 4.7. An input current pulse shortly after the reset at time evokes a postsynaptic potential of much lower amplitude than an input current pulse that arrives much later. Fig. 4.7 qualitatively reproduces the membrane trajectory of motoneurons when stimulated by the same input pattern (Poliakov et al., 1996; Powers and Binder, 1996).
In this section, we show that the leaky integrate-and-fire neuron defined in Section 4.1 is a special case of the Spike Response Model. We consider an integrate-and-fire neuron driven by external current I^{ext} and postsynaptic current pulses (t - t_{j}^{(f)}). The potential u_{i} is thus given by
u(t) | = u_{r} exp - | (4.34) |
+ w_{ij}exp - (t - t_{j}^{(f)} - s) ds | ||
+ exp - I_{i}^{ext}(t - s) ds | ||
= (t - ) + w_{ij}(t - , t - t_{j}^{(f)}) + (t - , s) I_{i}^{ext}(t - s) ds , |
In order obtain an explicit expression for the kernel (4.36) we have to specify the time course of the postsynaptic current (s). Here, we take (s) as defined in (4.21), viz.,
(s) = exp - s/ (s) . | (4.38) |
We have seen above that the Spike Response Model contains the integrate-and-fire model as a special case. In this example, we show in addition that even a generalization of the integrate-and-fire model that has a time dependent membrane time constant can be described within the SRM framework.
To be specific, we consider an integrate-and-fire model with spike-time dependent time constant, i.e., with a membrane time constant that is a function of the time since the last postsynaptic spike,
The phenomenological neuron model SRM_{0} introduced in Chapter 1.3.1 is a special case of the Spike Response Model. In this section we review its relation to the SRM and the integrate-and-fire model.
A simplified version of the spike response model can be constructed by
neglecting the dependence of and upon the first
argument. We set
(s) | = | (, s) | |
(s) | = | (, s) |
The simplified model SRM_{0} defined in (4.42) with the kernel defined in (4.43) can be reinterpreted as a model with a dynamic threshold,
(t - ) = - (t - ) , | (4.44) |
The basic equation of the leaky integrate-and-fire model, Eq. (4.3), is a linear differential equation. However, because of the reset of the membrane potential after firing, the integration is not completely trivial. In fact, there are two different ways of proceeding with the integration of Eq. (4.3). In Section 4.2.2 we have treated the reset as a new initital condition and thereby constructed an exact mapping of the integrate-and-fire model to the Spike Response Model. We now turn to the second method and describe the reset as a current pulse. As we will see, the result is an approximative mapping to the simplified model SRM_{0}.
Let us consider a short current pulse I_{i}^{out} = - q (t) applied to the RC circuit of Fig. 4.1. It removes a charge q from the capacitor C and lowers the potential by an amount u = - q/C. Thus, a reset of the membrane potential from a value of u = to a new value u = u_{r} corresponds to an `output' current pulse which removes a charge q = C ( - u_{r}). The reset takes place every time when the neuron fires. The total reset current is therefore
(s) | = | - ( - u_{r}) exp - (s) , | (4.51) |
(s) | = | exp - (s - s') ds' (s) , | (4.52) |
(s) | = | exp - (s) . | (4.53) |
We note that, in contrast to Eq. (4.42), we still have on the right-hand side of Eq. (4.50) a sum over past spikes of neuron i. According to Eq. (4.51) the effect of the -kernel decays with a time constant . In realistic spike trains, the interval between two spikes is typically much longer than the membrane time constant . Hence the sum over the terms is usually dominated by the most recent firing time t_{i}^{(f)} < t of neuron i. We therefore truncate the sum ofer f and neglect the effect of earlier spikes,
A careful comparison of Eq. (4.51) with Eq. (4.35) shows that the kernel is different from the kernel derived previously for the exact mapping of the integrate-and-fire model to the full Spike Response Model. The difference is most easily seen if we set the reset potential to u_{r} = 0. While the kernel in Eq. (4.35) vanishes in this case, the kernel is nonzero. In fact, whereas in the full SRM the reset is taken care of by the definition of (t - , s) and (t - , s), the reset in the simplified model SRM_{0} is included in the kernel . The relation between the kernels of the simplified model SRM_{0} to that of the full model are discussed below in more detail.
If (s) is given by (4.21), then the integral on the right-hand side of (4.52) can be done and yields
What is the relation between the kernel derived in (4.36) and the introduced in (4.52)? We will show in this paragraph that
(s, t) | = exp - (t - t') dt' - exp - (t - t') dt' | |
= exp - (t - t') dt' | ||
- exp - exp - (t - t' - s) dt' . | (4.58) |
By a completely analogous sequence of transformations it is possible to show that
Similarly we can compare the kernel in (4.35) and the kernel defined in (4.51),
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