The models discussed in this chapter are point neurons, i.e., models that do not take into account the spatial structure of a real neuron. In Chapter 2 we have already seen that the electrical properties of dendritic trees can be described by compartmental models. In this section, we want to show that neurons with a linear dendritic tree and a voltage threshold for spike firing at the soma can be mapped, at least approximatively, to the Spike Response Model.
We study an integrate-and-fire model with a passive dendritic tree described by n compartments. Membrane resistance, core resistance, and capacity of compartment are denoted by R_{T}^{}, R_{L}^{}, and C^{}, respectively. The longitudinal core resistance between compartment and a neighboring compartment is r^{} = (R_{L}^{} + R_{L}^{})/2; cf. Fig. . Compartment = 1 represents the soma and is equipped with a simple mechanism for spike generation, i.e., with a threshold criterion as in the standard integrate-and-fire model. The remaining dendritic compartments ( 2n) are passive.
Each compartment 1n of neuron i may receive input I_{i}^{}(t) from presynaptic neurons. As a result of spike generation, there is an additional reset current (t) at the soma. The membrane potential V_{i}^{} of compartment is given by
Equation (4.81) is a system of linear differential equations if the external input current is independent of the membrane potential. The solution of Eq. (4.81) can thus be formulated by means of Green's functions G_{i}^{}(s) that describe the impact of an current pulse injected in compartment on the membrane potential of compartment . The solution is of the form
We consider a network made up of a set of neurons described by Eq. () and a simple threshold criterion for generating spikes. We assume that each spike t_{j}^{(f)} of a presynaptic neuron j evokes, for t > t_{j}^{(f)}, a synaptic current pulse (t - t_{j}^{(f)}) into the postsynaptic neuron i; cf. Eq. (4.19). The voltage dependence of the synaptic input is thus neglected and the term (u_{i} - E_{syn}) in Eq. (4.20) is replaced by a constant. The actual amplitude of the current pulse depends on the strength w_{ij} of the synapse that connects neuron j to neuron i. The total input to compartment of neuron i is thus
In the following we assume that spikes are generated at the soma in the manner of the integrate-and-fire model. That is to say, a spike is triggered as soon as the somatic membrane potential reaches the firing threshold, . After each spike the somatic membrane potential is reset to V_{i}^{1} = u_{r} < . This is equivalent to a current pulse
Using the above specializations for the synaptic input current and the somatic reset current the membrane potential (4.82) of compartment in neuron i can be rewritten as
V_{i}^{}(t) = (t - t_{i}^{(f)}) + w_{ij}(t - t_{j}^{(f)}). | (4.86) |
The triggering of action potentials depends on the somatic membrane potential only. We define u_{i} = V_{i}^{1}, (s) = (s) and, for j , we set = . This yields
We illustrate the Spike Response method by a simple model with two compartments and a reset mechanism at the soma (Rospars and Lansky, 1993). The two compartments are characterized by a somatic capacitance C^{1} and a dendritic capacitance C^{2} = a C^{1}. The membrane time constant is = R^{1} C^{1} = R^{2} C^{2} and the longitudinal time constant = r^{12} C^{1} C^{2}/(C^{1} + C^{2}). The neuron fires, if V^{1}(t) = . After each firing the somatic potential is reset to u_{r}. This is equivalent to a current pulse
In the previous subsection we had to neglect the effect of spikes t_{i}^{(f)} (except that of the most recent one) on the somatic membrane potential of the neuron i itself in order to map Eq. (4.82) to the Spike Response Model. We can do better if we allow that the response kernels depend explicitly on the last firing time of the presynaptic neuron. This alternative treatment is an extension of the approach that has already been discussed in Section 4.2.2 in the context of a single-compartment integrate-and-fire model.
In order to account for the renewal property of the Spike Response Model we should solve Eq. (4.81) with initial conditions stated at the last presynaptic firing time . Unfortunately, the set of available initial conditions at is incomplete because only the somatic membrane potential equals u_{r} immediately after t = . For the membrane potential of the remaining compartments we have to use initial conditions at t = - , but we can use a short-term memory approximation and neglect indirect effects from earlier spikes on the present value of the somatic membrane potential.
We start with Eq. (4.82) and split the integration over s at s = into two parts,
V_{i}^{1}(t) | = ds G_{i}^{1}(t - s)I_{i}^{}(s) - (s) | |
+ ds G_{i}^{1}(t - s) I_{i}^{}(s) . | (4.94) |
With G_{i}^{1}(t - s) = G_{i}^{1}(t - ) G_{i}^{}( - s), which is a general property of Green's functions, we obtain
V_{i}^{1}(t) | = G_{i}^{1}(t - ) ds G_{i}^{}( - s)I_{i}^{}(s) - (s) | |
+ ds G_{i}^{1}(t - s) I_{i}^{}(s) . | (4.95) |
V_{i}^{1}( +0) = ds G_{i}^{1}( - s)I_{i}^{}(s) - (s) = u_{r} , | (4.96) |
V_{i}^{1}(t) | = G_{i}^{11}(t - ) u_{r} | |
+ G_{i}^{1}(t - )ds G_{i}^{}( - s)I_{i}^{}(s) - (s) | ||
+ ds G_{i}^{1}(t - s) I_{i}^{}(s) . | (4.97) |
V_{i}^{1}(t) | = G_{i}^{11}(t - ) u_{r} | |
+ G_{i}^{1}(t - )ds G_{i}^{}( - s) I_{i}^{}(s) | ||
+ ds G_{i}^{1}(t - s) I_{i}^{}(s) | ||
+ ( - u_{r})G_{i}^{1}(t - ) G_{i}^{1}( - t_{i}^{(f)}) . | (4.98) |
(r, s) = | (4.99) |
V_{i}^{1}(t) | = G_{i}^{11}(t - ) u_{r} | |
+ ds (t - , t - s) I_{i}^{}(s) | ||
+ ( - u_{r})G_{i}^{1}(t - ) G_{i}^{1}( - t_{i}^{(f)}) . | (4.100) |
If we neglect the last term in Eq. (4.100), that is, if we neglect any indirect effects of previous action potentials on the somatic membrane potential, then Eq. (4.100) can be mapped on the Spike Response Model (4.24) by introducing kernels
(r, s) = dt' (r, t') (t' - s) , | (4.101) |
(s) = G_{i}^{11}(s) u_{r} . | (4.102) |
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