# Spike-response model

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Curator: Wulfram Gerstner

The Spike Response Model is a generalization of the leaky integrate-and-fire model and gives a simple description of action potential generation in neurons. Just as in the integrate-and-fire model, action potentials are generated when the voltage passes a threshold from below. In contrast to the leaky integrate-and-fire model, the spike response model includes refractoriness. A notational difference is that integrate-and-fire models are formulated using differential equations for the voltage, whereas the Spike Response Model is formulated using filters. Figure 1: The Spike Response Model contains a filter $$\kappa$$ which describes the voltage response (red, zoomed region) to an incoming pulse (input current shown in blue), a function$$\eta$$ which describes the form of an action potential (green), and a dynamic threshold which increases after a spike has been triggered (dashed line).

## Mathematical formulation

The membrane potential in the spike response model is given by $\tag{1} u(t) = \eta(t-\hat{t}) + \int_0^\infty \kappa(t-\hat{t},s) I(t-s) ds$

where $$\hat{t}$$ is the firing time of the last spike of the neuron, $$\eta$$ describes the form of the action potential and its spike after-potential, $$\kappa$$ the linear response to an input pulse and $$I(t)$$ a stimulating current. The next spike occurs if the membrane potential $$u$$ hits a threshold $$\theta(t-\hat{t})$$ from below in which case $$\hat{t}$$ is updated.

• The name spike response model stems from the fact that $$\kappa$$ describes the response of the neuron to an incoming short pulse (such as a spike arriving from another neuron) and $$\eta$$ describes the response of the membrane to its own spike. The functions $$\eta$$ and $$\kappa$$ are also called kernels, $$\kappa$$ is also called the linear filter of the membrane.
• From the mathematical point of view it is not necessary to keep both a dynamic threshold and the spike shape $$\eta\ ,$$ since only the difference $$x=u - \theta$$ between the membrane potential and the threshold matters for the spike dynamics.

## Main features and examples

• The threshold $$\theta$$ is not fixed but depends on the time since the last spike. Typically the threshold is higher immediately after a spike and decays then back to its resting value.
• The spike shape $$\eta$$ is a function of the time since the last spike. It can describe a depolarizing, hyperpolarizing, or resonating spike-after potential (see Figure 3 and Figure 2). Figure 4: Example of an impulse response current $$\kappa(dt,s)$$ as a function of s, extracted from data for three different values of the time dt that has passed since the last spike. Figure 5: Example of an impulse response current $$\kappa$$ with damped oscillations, extracted from the Hodgkin-Huxley model. The graph shows $$\kappa(dt,s)$$ as a function of s for different choices of the time dt that has passed between the last postsynaptic spike and presynaptic spike arrival.
• The responsiveness $$\kappa$$ to an input pulse depends on the time since the last spike, because typically the effective membrane time constant after a spike is shorter, since many ion channels are open.
• The time course of the response $$\kappa$$ can include a single exponential, combinations of exponentials with different time constants, or resonating behavior in form of a delayed oscillation. This is the case if the standard Hodgkin-Huxley model is approximated by the Spike Response Model.
• Refractoriness can be modelled as a combination of increased threshold, hyperpolarizing afterpotential, and reduced responsiveness after a spike, as observed in real neurons (Badel et al., 2008).

## Fits to experimental data

The Spike Response Model can be fitted to experimental data where a neuron is stimulated by a rapidly varying time dependent current or conductance, see Figure 6 To do so, use the following steps. Figure 6: The same time-dependent input (left) is given to the Spike Response Model (top) and a real neuron (bottom). Comparison of the voltage (right) shows that the Spike Response Model follows the subthreshold membrane potential and predicts spike times.
• subtract from the experimental data the resting potential. This gives a normalized voltage trace $$u(t)\ .$$
• align spikes and determine the mean shape of the spike and spike-afterpotential. This gives $$\eta(t-\hat{t})\ .$$
• remove spikes and calculate the subthreshold membrane potential $$y(t) = u(t) - \eta(t-\hat{t})$$
• determine the best linear filter to approximate the subthreshold potential $$y(t)$$ by $$\int_0^\infty \kappa(t-\hat{t},s) I^{\rm ext}(t-s) ds\ .$$
• optimize the threshold $$\theta$$ so as to get the correct mean firing rate of the neuron.

The above parameter extraction procedure yields a Spike Response Model that fits experimental data to a high degree of accuracy and predicts a large fraction of spikes with a precision of +/-2ms (Jolivet et al., 2006). The same procedure has also been used to approximate detailed neuron models of the Hodgkin-Huxley type by the Spike Response Model (Kistler et al., 1997, Jolivet et al., 2004).

In a public competition of spike-time prediction under random conductance injection, the Spike Response Model was in the group of winning models (Jolivet et al., 2008) whereas a standard leaky integrate-and-fire model performed significantly worse. These and other results show that inclusion of refractory properties is important. Refractory properties can be measured directly and manifest themselves in the time course of the spike afterpotenial $$\eta(t-\hat{t})\ ,$$ by a reduction in the effective membrane time constant influencing $$\kappa(t-\hat{t})\ ,$$ and by an increase of the threshold $$\theta$$ after a spike (Badel et al., 2008).

## Special cases and variants of the Spike Response Model

### Leaky Integrate-and-fire model

The leaky integrate and fire model is a special case of the Spike Response Model. To see this, take the differential equation of the leaky integrate-and-fire model ${dv \over dt} = -{{v - v_{eq}} \over {\tau}} + I(t)$ where $$v$$ is the membrane potential, $$v_{eq}$$ is the equilibrium potential and $$\tau = RC$$ is the membrane time constant. Integration of the differential equation for arbitrary input $$I(t)\ ,$$ yields $\tag{2} u(t) = \eta(t-\hat{t}) + \int_0^\infty \Theta(t-\hat{t}-s) \exp(-s/\tau)\, I(t-s) ds$

where $$u = v-v_{eq}$$ and $$\eta(t-\hat{t})= (v_{reset} - v_{eq}) \, \exp[-(t-\hat{t})/\tau]$$ comes from the reset after each spike. This is a special case of the Spike Response Model defined in Eq. (1) with $$\kappa(t-\hat{t},s) = \Theta(t-\hat{t}-s) \exp(-s/\tau)$$ where $$\Theta$$ denotes the Heaviside step function.

### Leaky Integrate-and-fire model with time-dependent time constant

The effective time constant of a neuron is different immediately after an action potential. Similarly, the momentary equilibrium potential is different and varies as a function of the time since the last spike. Hence we can write an integrate-and-fire model with time-dependent parameters ${dv\over dt} = -{{v-v_{eq}(t-\hat{t})}\over{\tau(t-\hat{t})}} + I(t)$ Integration of this equation gives a special case of the Spike Response Model (1).

### Spike input and synaptic response kernel $$\epsilon$$

In the case of input generated by synaptic current pulses caused by presynaptic spike arrival, the Spike Response Model can be written as $\tag{3} u(t) = \eta(t-\hat{t}) + \sum_j \sum_f w_j \epsilon(t-\hat{t},t-t_j^f)$

where $$w_j$$ is the weight of synapse $$j\ ,$$ $$t_j^f$$ the arrival time of the $$f$$ th spike at synapse $$j\ ,$$ and $$\epsilon$$ the time course of a postsynaptic potential caused by spike arrival.

To see the connection with Eq. (1), suppose that the input to the Spike Response Model consists not of an imposed current, but of synaptic input currents of amplitude $$w_j$$ and time course $$\alpha(t-t_j^f)$$ where $$t_j^f$$ is the spike arrival time at synapse $$j\ ,$$ that is, $$I(t) = \sum_j \sum_f w_j \, \alpha(t-t_j^f)\ .$$ Convolution of the kernel $$\kappa$$ with the current $$\alpha$$ yields the postsynaptic potential $$\epsilon\ .$$

### Spike Response Model SRM$$_0$$ Figure 7: The Spike Response Model SRM$$_0$$ has a filter $$\kappa$$ which describes the voltage response (red, zoomed region) to an incoming pulse (input current shown in blue). In contrast to the full Spike Response Model, this filter does not depend on the time since the last spike. Compared to Figure 1, this leads to a less pronounced reset after each spike. As before, the function$$\eta$$ describes the form of an action potential (green), and a dynamic threshold increases after a spike has been triggered (dashed line).

SRM$$_0$$ is a simplified version of the Spike Response Model. It does not include a dependence of the response kernel $$\kappa$$ upon the time since the last spike. Hence we have $\tag{4} u(t) = \eta(t-\hat{t}) + \int \kappa(s) I^{\rm ext}(t-s) ds$

instead of (1). The threshold can be dynamic as before. The variant SRM$$_0$$ is easier to fit to experimental data than the full Spike Response Model, since it needs less data (Jolivet et al. 2006).

### Cumulative Spike Response Model: bursting and adaptation

In some papers (Gerstner et al. 1996) the term Spike Response Model was used for a model where refractoriness and adaptation were modeled by the combined effects of the spike after potentials of several previous spikes, rather than only the most recent spike. Hence equation (1) is replaced by $\tag{5} u(t) = \sum_{t^k} \eta(t-t^k) + \int \kappa(t-\hat{t},s) I^{\rm ext}(t-s) ds$

where $$t^k<t$$ denotes previous moments of spike firing. Similarly, the threshold increases .

The reason for keeping in the normal Spike Response Model only the effect of the most recent spike is that a closed-form mathematical analysis of large networks in the form of mean-field equations (Gerstner 2000, Gerstner and van Hemmen 1992) is much easier in the standard Spike Response Model than in the cumulative Spike Response Model. The advantage of the cumulative model is that it accounts for adaptation and bursting (Gerstner and van Hemmen 1992, Gerstner et al. 1996).

### Resonate-and-fire model

The resonate-and-fire model (Izhikevich 2001, Richardson et al. 2003) is a special case of the Cumulative Spike Response Model. The resonate and fire model consists of two linear equations: ${du \over dt} = -{{v - v_{eq}} \over {\tau}} - c w + I(t)$ $\tau_w {dw \over dt} = a (v - v_{eq}) - w$ where $$w$$ is a second variable that summarizes the effect of subthreshold membrane current. The voltage variable $$v$$ is reset whenever it reaches a firing threshold. Integration of these linear equations gives a solution of the form (1) with a specific choice for the filters $$\eta$$ and $$\kappa\ .$$

### Spike Response Model with a cumulative dynamic threshold Figure 8: In the Spike Response Model with cumulative threshold, the dynamic threshold increases after each spike by a fixed amount so that the effects of several spikes accumulate (dashed line). This is in contrast to the normal spike response model where the threshold always restarts at the same value (see Fig 1).

The threshold $$\theta$$ is calculated as $\tag{6} \theta(t) = \theta_0 + \sum_{t^k} \vartheta(t-t^k)$

where $$t^k<t$$ denotes previous moments of spike firing, $$\theta_0$$ is the value of the threshold at rest, and $$\vartheta(t-t^k)$$ describes the effect of a spike at time $$t^k$$ on the value of the threshold at time $$t\ .$$ The difference to the standard form of the spike response model is that now the value of the threshold depends on all previous spikes, not only the most recent one. Having a cumulative threshold is one possible way to incorporate adaptation. Furthermore, a spike response model with cumulative threshold makes it possible to have a single model with a fixed set of parameters that fits experimental data across a broad range of firing rates (Jolivet et al. 2006).

## Noise in the Spike Response Model

Noise can be included into the Spike Response Model by replacing the strict threshold criterion $$u(t) = \theta$$ by a stochastic process. The probability $$P$$ of firing a spike within a very short time $$\Delta t$$ is $$P=\rho(t)\,\Delta t$$ where the instantaneous firing rate or firing intensity $$\rho(t)$$ is a function of the momentary difference between the membrane potential $$u(t)$$ and the threshold $$\theta(t),$$ $\tag{SRMcum-eq:label exists!} \rho = f(u-\theta).$

This noise model has been called stochastic threshold or escape noise (Gerstner and Kistler 2002). Typical functional forms for $$f$$ could be an exponential function (Jolivet et al. 2006) or a rectified linear function. A saturating form for $$f$$ is not a good choice, because it introduces a finite response time even for strong input well above threshold (Gerstner and Kistler 2002).

The advantage of the escape noise model in combination with a Spike Response Model is that interspike interval distributions can be expressed analytically for arbitrary time-dependent input. Given a spike at time $$\hat{t}$$ and an input $$I(t)\ ,$$ the next spike will occur at time $$t$$ with probability density $\tag{SRMcum-eq:label exists!} P(t|\hat{t}) = \rho(t) \, \exp[ \int_\hat{t}^t -\rho(t')\, dt']$

From the point of view of stochastic processes, this is a time-dependent (inhomogeneous) version of a renewal model.

Because stochastic spike arrival with time-dependent input rates can be well represented by an equivalent escape noise model (Plesser and Gerstner 2000), one can calculate interspike interval distributions even for cases where the first-passage time problem of the leaky integrate-and-fire model with diffusive noise cannot be solved analytically. The Spike Response Model with escape noise has turned out to be useful for description of experimental data (Jolivet et al. 2006, Pillow et al. 2008).

## Effects not captured by a Spike Response Model

The Spike Response Model is not suited to describe the following effects:

• Pharmacological blocking of ion channels.

The biophysics of the neuronal membrane is not described explicitly in the Spike Response Model. Instead, the combined effects of several ion channel are captured phenomenologically in the spike shape function $$\eta$$ and the filter $$\kappa\ .$$ Therefore the model cannot make predictions about blocking of individual ion channels. A Hodgkin-Huxley model is better suited to describe the effects of individual channels.

• Delayed spike initiation.

For isolated input pulses, type I neuron models exhibit action potentials of standard shape, but different delay depending on the amplitude of the input pulse. The Spike Response Model cannot capture these effects, because of the strict threshold criterion. A quadratic (Latham et al. 2000) or an exponential integrate-and-fire model (Fourcaud-Trocme et al. 2003) is better suited to describe these effects.

• Dependence of the threshold upon the input.

The critical voltage for spike initiation with very slow ramp currents or constant currents (rheobase threshold) is different in real neurons from the critical voltage for spike initiation with short current pulses. The Spike Response Model cannot capture these effects because of the strict threshold criterion. An adaptive exponential integrate-and-fire (Brette and Gerstner, 2006) model is better suited to describe these effects.

## History

Simple spiking neurons with a formal firing threshold that potentially increases after each spike have a long history (e.g., Lapique 1907, Hill 1936, Weiss 1966). Models closely related to the Spike Response Model have been used for quantitative prediction of neural spike trains by several groups (Brillinger 1992, Keat et al. 2001, Jolivet et al. 2004, Carandini 2007, Pillow 2008). The term Spike Response Model was introduced around 1993 (Gerstner et al., 1993; Gerstner 1995).

## Summary

A major advantage of formal spiking neurons such as the Spike Response Model is their simplicity which has several important consequences.

• It is possible to simulate neural networks with a large number of neurons at a reasonable numerical cost.
• Network properties such as the mean firing rate of neurons in a network of connected spiking units can be studied analytically using tools from mathematical probability theory, statistical physics, and bifurcation theory.
• Questions of neural coding can be addressed in a transparent fashion.
• Simple spiking neuron models can be fitted to experimental data.
• Compared to standard leaky integrate-and-fire models the Spike Response Model allows to cover refractoriness.
• The spike response model is the most general model that combines linear filtering with a strict threshold. Hence it incorporates several spiking neuron models as special cases.