So far, we have described neuronal dynamics in terms of systems of differential equations. There is another approach that was introduced in Section 1.3.5 as the ‘filter picture’. In this picture, the parameters of the model are replaced by (parametric) functions of time, generically called ‘filters’. The neuron model is therefore interpreted in terms of a membrane filter as well as a function describing the shape of the spike (Fig. 6.11) and, potentially, also a function for the time course of the threshold. Together, these three functions establish the Spike Response Model (SRM).
The Spike Response Model is – just like the nonlinear integrate-and-fire models in Chapter 5 or the AdEx in Section 6.1 – a generalization of the leaky integrate-and-fire model. In contrast to nonlinear integrate-and-fire models, the SRM has no ‘intrinsic’ firing threshold but only the sharp numerical threshold for reset. If the nonlinear function of the AdEx is fitted to experimental data, the transition between the linear subthreshold and superthreshold behavior is found to be rather abrupt, so that the nonlinear transition is, for most neurons, well approximated by a sharp threshold (see Fig. 5.3). Therefore, in the SRM, we work with a sharp threshold combined with a linear voltage equation.
While the SRM is therefore somewhat simpler than other models on the level of the spike generation mechanism, the subthreshold behavior of the SRM is richer than that of the integrate-and-fire model discussed so far and can account for various aspects of refractoriness and adaptation. In fact, the SRM combines the most general linear model with a sharp threshold.
It turns out that the integral formulation of the SRM is very useful for data fitting and also the starting point for the Generalized Linear Models in Chapter 9 and 10. Despite the apparent differences between integrate-and-fire models and SRM, the leaky integrate-and-fire model, with or without adaptation variables, is a special case of the SRM. The relation of the SRM to integrate-and-fire models is the topic of Section 6.4.3 – 6.4.4. We now start with a detailed explanation.
In the framework of the Spike Response Model (SRM) the state of a neuron is described by a single variable which we interpret as the membrane potential. In the absence of input, the variable is at its resting value, . A short current pulse will perturb and it takes some time before returns to rest (Fig. 6.11). The function describes the time course of the voltage response to a short current pulse at time . Because the subthreshold behavior of the membrane potential is taken as linear, the voltage response to an arbitrary time-dependent stimulating current is given by the integral .
Spike firing is defined by a threshold process. If the membrane potential 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 that the neuron has fired some earlier spikes at times . The evolution of is given by
The sum runs over all past firing times with of the neuron under consideration. Introducing the spike train , Eq. (6.27) can be also written as a convolution
In contrast to the leaky integrate-and-fire neuron discussed in Chapter 1 the threshold is not fixed, but time-dependent
Firing occurs whenever the membrane potential reaches the dynamic threshold from below
The kernel 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”).
The kernel describes the standard form of an action potential of neuron including the negative overshoot which typically follows a spike (the spike afterpotential). Graphically speaking, a contribution is ‘pasted in’ each time the membrane potential reaches the threshold (Fig. 6.12A). 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 .
In a simplified model, the form of the action potential may therefore be neglected as long as we keep track of the firing times . 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
with a parameter . The spike after-potential decays back to zero with a recovery time constant . The leaky integrate-and-fire model is in fact a special case of the SRM, with parameter and .
In this section, we show that the leaky integrate-and-fire neuron with adaptation defined above in Eq. (6.7) and (6.8) is a special case of the Spike Response Model. Let us recall that the leaky integrate-and-fire model follows the equation of a linear circuit with resistance and capacity
where is the time constant, the leak reversal potential, are adaptation variables, and is the input current to neuron . At each firing time
the voltage is reset to a value . At the same time, the adaptation variables are increased by an amount
The equations of the adaptive leaky integrate-and-fire model, Eqs. (6.35) and (6.37) can be classified as linear differential equations. 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. The first method is to treat the reset after each firing as a new initial condition - this is the procedure typically chosen for a numerical integration of the model. Here we follow a different path and describe the reset as a current pulse. As we will see, the result enables a mapping of the leaky integrate-and-fire model to the SRM.
Let us consider a short current pulse applied to the circuit. It removes a charge from the capacitor and lowers the potential by an amount . Thus, a reset of the membrane potential from a value of to a new value corresponds to an ‘output’ current pulse which removes a charge . The reset takes place every time when the neuron fires. The total reset current is therefore
Since Eqs. (6.39) and (6.40) define a system of linear equations, we can integrate each term separately and superimpose the result at the end. To perform the integration, we proceed in three steps. First, we shift the voltage so as to set the equilibrium potential to zero. Second, we calculate the eigenvalues and eigenvectors of the ‘free’ equations in the absence of input (and therefore no spikes). If there are adaptation variables, we have a total of eigenvalues which we label as . The associated eigenvectors are with components . Third, we express the response to an impulse in the voltage (no perturbation in the adaptation variables) in terms of the eigenvectors: . Finally, we express the pulse caused by a reset of voltage and adaptation variables in terms of the eigenvectors .
The response to the reset pulses yields the kernel while the response to voltage pulses yields the filter of the SRM
As usual, denotes the Heaviside step function.
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 3 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 to the Spike Response Model.
We study an integrate-and-fire model with a passive dendritic tree described by compartments. Membrane resistance, core resistance, and capacity of compartment are denoted by , , and , respectively. The longitudinal core resistance between compartment and a neighboring compartment is ; cf. Fig. 3.8. Compartment 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 () are passive.
Each compartment of neuron may receive input from presynaptic neurons. As a result of spike generation, there is an additional reset current at the soma. The membrane potential of compartment is given by
where the sum runs over all neighbors of compartment . The Kronecker symbol equals unity if the upper indices are equal; otherwise, it is zero. The subscript is the index of the neuron; the upper indices or refer to compartments. Below we will identify the somatic voltage with the potential of the Spike Response Model.
Equation (6.45) is a system of linear differential equations if the external input current is independent of the membrane potential. The solution of Eq. (6.45) can thus be formulated by means of Green’s functions 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. (6.45) and a simple threshold criterion for generating spikes. We assume that each spike of a presynaptic neuron evokes, for , a synaptic current pulse into the postsynaptic neuron . The actual amplitude of the current pulse depends on the strength of the synapse that connects neuron to neuron . The total input to compartment of neuron is thus
Here, denotes the set of all neurons that have a synapse with compartment of neuron . The firing times of neuron are denoted by .
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 . This is equivalent to a current pulse
so that the overall current due to the firing of action potentials at the soma of neuron amounts to
Using the above specializations for the synaptic input current and the somatic reset current the membrane potential (6.46) of compartment in neuron can be rewritten as
The kernel describes the effect of a presynaptic action potential arriving at compartment on the membrane potential of compartment . Similarly, describes the response of compartment to an action potential generated at the soma.
The triggering of action potentials depends on the somatic membrane potential only. We define , and, for , we set . This yields the equation of the SRM
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