In the SRM, the membrane voltage V is described as a linear sum of the postsynaptic potentials (PSPs) caused by spike arrivals to which the effects of refractoriness and adaptation are added.
The SRM is flexible enough to account for a variety of neuronal firing pattern in response to step current input.
[2] The SRM has also been used in the theory of computation to quantify the capacity of spiking neural networks;[3] and in the neurosciences to predict the subthreshold voltage and the firing times of cortical neurons during stimulation with a time-dependent current stimulation.
[4] The name Spike Response Model points to the property that the two important filters
[2][8][9] In the SRM, at each moment in time t, a spike can be generated stochastically with instantaneous stochastic intensity or 'escape function'[8][9][5] that depends on the momentary difference between the membrane voltage V(t) and the dynamic threshold
is a time constant that describes how quickly a spike is fired once the membrane potential reaches the threshold and
The escape rate process via a soft threshold is reviewed in Chapter 9 of the textbook Neuronal Dynamics.
describe the amplitude and time course of an excitatory or inhibitory postsynaptic potential (PSP) caused by the spike
, a spike is generated with probability that depends on the momentary difference between the membrane voltage V and the dynamic threshold
and rewrite the membrane potential as[5] In this notation, the refractory kernel
can be interpreted as linear response filters applied to the binary spike trains
Since the formulation as SRM provides an explicit expression for the membrane voltage (without the detour via a differential equations), SRMs have been the dominant mathematical model in a formal theory of computation with spiking neurons.
[12][13][3] The SRM with dynamic threshold has been used to predict the firing time of cortical neurons with a precision of a few milliseconds.
[14][15] Thanks to the convexity properties of the likelihood in Generalized Linear Models,[6][7] parameter extraction is efficient.
[5][11] The SRM network[5] which stored a finite number of stationary patterns as attractors using a Hopfield-type connectivity matrix[17] was one of the first examples of attractor networks with spiking neurons.
[5][10] Because the refractory kernel may include a time scale slower than that of the membrane potential, the population equations for SRM neurons provide powerful alternatives[22][21][23] to the more broadly used partial differential equations for the 'membrane potential density'.
[19][24][25] Reviews of the population activity equation based on refractory densities can be found in[23] as well in Chapter 14 of the textbook Neuronal Dynamics.
A network SRM neurons has stored attractors that form reliable spatio-temporal spike patterns[1] (also known as synfire chains[26]) example of temporal coding for stationary inputs.
Moreover, the population activity equations for SRM exhibit temporally precise transients after a stimulus switch, indicating reliable spike firing.
[10] The Spike Response Model has been introduced in a series of papers between 1991[11] and 2000.
[28][29][30][31] An important variant of the model is SRM0[10] which is related to time-dependent nonlinear renewal theory.
The main difference to the voltage equation of the SRM introduced above is that in the term containing the refractory kernel
[29][30][31] The equations of the SRM as introduced above are equivalent to Generalized Linear Models in neuroscience (GLM).
[6][7] In the neuroscience, GLMs have been introduced as an extension of the Linear-Nonlinear-Poisson model (LNP) by adding self-interaction of an output spike with the internal state of the neuron[6][7] (therefore also called 'Recursive LNP').
The GLM framework enables to formulate a maximum likelihood approach[33] applied to the likelihood of an observed spike train under the assumption that an SRM could have generated the spike train.
[8] Despite the mathematical equivalence there is a conceptual difference in interpretation: in the SRM the variable V is interpreted as membrane voltage whereas in the recursive LNP it is a 'hidden' variable to which no meaning is assigned.
The SRM interpretation is useful if measurements of subthreshold voltage are available[4][14][15] whereas the recursive LNP is useful in systems neuroscience where spikes (in response to sensory stimulation) are recorded extracellulary without access to the subthreshold voltage.
[6][7] A leaky integrate-and-fire neuron with spike-triggered adaptation has a subthreshold membrane potential generated by the following differential equations where
is the membrane time constant and wk is an adaptation current number, with index k, Erest is the resting potential and tf is the firing time of the neuron and the Greek delta denotes the Dirac delta function.
In the absence of adaptation currents, we retrieve the standard LIF model which is equivalent to a refractory kernel