6 Adaptation and Firing Patterns

6.4 Spike response model (SRM)

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 We now start with a detailed explanation.

6.4.1 Definition of the SRM

Fig. 6.11: Spike Response Model (SRM). Input current I(t)I(t) is filtered with a filter κ(s)\kappa(s) and yields the input potential h(t)=0κ(s)I(t-s)dsh(t)=\int_{0}^{\infty}\kappa(s)I(t-s){\text{d}}s. Firing occurs if the membrane potential uu reaches the threshold ϑ\vartheta. Spikes S(t)=fδ(t-t(f))S(t)=\sum_{f}\delta(t-t^{(f)}) are fed back into the threshold process in two distinct ways. Each spike causes an increase θ1\theta_{1} of the threshold: ϑ(t)=ϑ0+0θ1(s)S(t-s)ds\vartheta(t)=\vartheta_{0}+\int_{0}^{\infty}\theta_{1}(s)S(t-s){\text{d}}s. Moreover, each spike generates a voltage contribution η\eta to the membrane potential: u(t)=h(t)+0η(s)S(t-s)dsu(t)=h(t)+\int_{0}^{\infty}\eta(s)S(t-s){\text{d}}s, where η\eta captures the time course of the action potential and the spike-afterpotential; schematic figure.

In the framework of the Spike Response Model (SRM) the state of a neuron is described by a single variable uu which we interpret as the membrane potential. In the absence of input, the variable uu is at its resting value, urestu_{\rm rest}. A short current pulse will perturb uu and it takes some time before uu returns to rest (Fig. 6.11). The function κ(s)\kappa(s) describes the time course of the voltage response to a short current pulse at time s=0s=0. Because the subthreshold behavior of the membrane potential is taken as linear, the voltage response hh to an arbitrary time-dependent stimulating current Iext(t)I^{\rm ext}(t^{\prime}) is given by the integral h(t)=0κ(s)Iext(t-s)dsh(t)=\int_{0}^{\infty}\kappa(s)\,I^{\rm ext}(t-s)\,{\text{d}}s.

Spike firing is defined by a threshold process. If the membrane potential reaches the threshold ϑ\vartheta, an output spike is triggered. The form of the action potential and the after-potential is described by a function η\eta. Let us suppose that the neuron has fired some earlier spikes at times t(f)<tt^{(f)}<t. The evolution of uu is given by

u(t)\displaystyle u(t) =\displaystyle= fη(t-t(f))+0κ(s)Iext(t-s)ds+urest\displaystyle\sum_{f}\eta(t-t^{(f)})+\int_{0}^{\infty}\kappa(s)\,I^{\rm ext}(t% -s)\,{\text{d}}s+u_{\rm rest} (6.27)

The sum runs over all past firing times t(f)t^{(f)} with f=1,2,3,f=1,2,3,\dots of the neuron under consideration. Introducing the spike train S(t)=fδ(t-t(f))S(t)=\sum_{f}\delta(t-t^{(f)}), Eq. (6.27) can be also written as a convolution

u(t)\displaystyle u(t) =\displaystyle= 0η(s)S(t-s)ds+0κ(s)Iext(t-s)ds+urest\displaystyle\int_{0}^{\infty}\eta(s)S(t-s){\text{d}}s+\int_{0}^{\infty}\kappa% (s)\,I^{\rm ext}(t-s)\,{\text{d}}s+u_{\rm rest} (6.28)

In contrast to the leaky integrate-and-fire neuron discussed in Chapter 1 the threshold ϑ\vartheta is not fixed, but time-dependent

ϑϑ(t).\vartheta\quad\longrightarrow\quad{\vartheta}(t)\,. (6.29)

Firing occurs whenever the membrane potential uu reaches the dynamic threshold ϑ(t)\vartheta(t) from below

t=t(f)u(t)=ϑ(t) and d[u(t)-ϑ(t)]dt>0 .t=t^{(f)}\quad\Leftrightarrow\quad u(t)=\vartheta(t)\text{ and }{{\text{d}}[u(% t)-\vartheta(t)]\over{\text{d}}t}>0\,. (6.30)

Dynamic thresholds can be directly measured in experiments (162; 35; 338) and are a standard feature of phenomenological neuron models.

Example: Dynamic threshold - and how to get rid of it

A standard model of the dynamic threshold is

ϑ(t)=ϑ0+fθ1(t-t(f))=ϑ0+0θ1(s)S(t-s)   d   s,\vartheta(t)=\vartheta_{0}+\sum_{f}\theta_{1}(t-t^{(f)})=\vartheta_{0}+\int_{0% }^{\infty}\theta_{1}(s)S(t-s){\text{d}}s\,, (6.31)

where ϑ0\vartheta_{0} is the ‘normal’ threshold of neuron ii in the absence of spiking. After each output spike, the firing threshold of the neuron is increased by an amount θ1(t-t(f))\theta_{1}(t-t^{(f)}) where t(f)<tt^{(f)}<t denote the firing times in the past. For example, during an absolute refractory period Δabs\Delta^{\rm abs}, we may set Δϑ{\Delta_{\vartheta}} for a few milliseconds to a large and positive value so as to avoid any firing and let it relax back to zero over the next few hundred milliseconds; cf. Fig. 6.12A.

From a formal point of view, there is no need to interpret the variable uu as the membrane potential. It is, for example, often convenient to transform the variable uu so as to remove the time-dependence of the threshold. In fact, a general Spike Response Model with arbitrary time-dependent threshold as in Eq. (6.31) can always be transformed into a Spike Response Model with fixed threshold ϑ0\vartheta_{0} by a change of variables

η(t-t(f))ηeff(t-t(f))=η(t-t(f))-θ1(t-t(f))\eta(t-t^{(f)})\longrightarrow\eta^{\rm eff}(t-t^{(f)})=\eta(t-t^{(f)})-\theta% _{1}(t-t^{(f)}) (6.32)

In other words, the dynamic threshold can be absorbed in the definition of the η\eta kernel. Note, however, that in this case η\eta can no longer be interpreted as the experimentally measured spike after-potential, but must be interpreted as an ‘effective’ spike after-potential.

The argument can also be turned the other way round, so as to remove the spike after-potential and only work with a dynamic threshold; see Fig. 6.12B. However, when an SRM is fitted to experimental data, it is convenient to separate the spike after-effects that are visible in the voltage trace (e.g., in the form of a hyperpolarizing spike-afterpotential, described by the kernel η\eta), from the spike after-effects caused by an increase in the threshold which can be observed only indirectly via the absence of spike firing. Whereas the prediction of spike times is insensitive to the relative contribution of η\eta and Δϑ\Delta_{\vartheta}, the prediction of the subthreshold voltage time course is not. Therefore, it is useful to explicitly work with two distinct adapatation mechanisms in the SRM (338).

Fig. 6.12: Spike afterpotential and dynamic threshold in the SRM. A. At time t(f)t^{(f)} a spike occurs because the membrane potential hits the threshold ϑ(t)\vartheta(t). The threshold jumps to a higher value (dashed line) and, at the same time, a contribution η(t-t(f))\eta(t-t^{(f)}) is added to the membrane potential, i.e., the spike and its spike-afterpotential are ‘pasted’ into the picture. If no further spikes are triggered, the threshold decays back to its resting value and the spike after-potential decays back to zero. The total membrane potential (thick solid line) after a spike is u(t)=h(t)+fη(t-t(f))u(t)=h(t)+\sum_{f}\eta(t-t^{(f)}) where h(t)h(t) is the input potential (thin dotted line). B. If the model is used to predict spike times, but not the membrane potential, the spike afterpotential η\eta can be integrated into the dynamic threshold so that u(t)=h(t)u(t)=h(t). At the moment of spiking the value of the threshold is increased, but the membrane potential is not affected (neither through reset, nor spike afterpotential).

6.4.2 Interpretation of η\eta and κ\kappa

So far Eq. (6.27) in combination with the threshold condition (6.30) defines a mathematical model. Can we give a biological interpretation of the terms?

The kernel κ(s)\kappa(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”).

The kernel η\eta describes the standard form of an action potential of neuron ii including the negative overshoot which typically follows a spike (the spike afterpotential). Graphically speaking, a contribution η\eta is ‘pasted in’ each time the membrane potential reaches the threshold ϑ\vartheta (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 t(f)t^{(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(f)t^{(f)}. The kernel η\eta describes then simply the ‘reset’ of the membrane potential to a lower value after the spike at t(f)t^{(f)} just like in the integrate-and-fire model

η(t-t(f))=-η0exp(-t-t(f)τrecov),\eta(t-t^{(f)})=-\eta_{0}\,\exp\left(-{t-t^{(f)}\over\tau_{\rm recov}}\right)\,, (6.33)

with a parameter η0>0\eta_{0}>0. The spike after-potential decays back to zero with a recovery time constant τrecov\tau_{\rm recov}. The leaky integrate-and-fire model is in fact a special case of the SRM, with parameter η0=(ϑ-ur)\eta_{0}=(\vartheta-u_{r}) and τrecov=τm\tau_{\rm recov}=\tau_{m}.

Example: Refractoriness

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. In Figure 5.5 (see Chapter 5) we have already seen that refractoriness shows up as increased firing threshold and increased conductance immediately after a spike.

Absolute refractoriness can be incorporated in the SRM by setting the dynamic threshold during a time Δabs\Delta^{\rm abs} to an extremely high value that cannot be attained.

Relative refractoriness can be mimicked in various ways. First, after a spike the firing threshold returns only slowly back to its normal value (increase in firing threshold). Second, after the spike the membrane potential, and hence η\eta, passes through a regime of hyperpolarization (spike 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. In fact, this is equivalent to a transient increase of the firing threshold (see above).

Third, the responsiveness of the neuron is reduced immediately after a spike. In the SRM we can model the reduced responsiveness by making the shape of ϵ\epsilon and κ\kappa depend on the time since the last spike timing t^\hat{t}.

We label output spike such that the most recent one receives the label t1t^{1} (i.e. t>t1>t2>t3t>t^{1}>t^{2}>t^{3}\dots). This means that after each firing event, output spikes need to be relabeled. The advantage, however, is that the last output spike always keeps the label t1t^{1}. For simplicity, we often write t^\hat{t} instead of t1t^{1} to denote the most recent spike.

With this notation, a slightly more general version of the Spike Response Model is

u(t)\displaystyle u(t) =\displaystyle= fη(t-t(f))+0κ(t-t^,s)Iext(t-s)   d   s+urest.\displaystyle\sum_{f}\eta(t-t^{(f)})+\int_{0}^{\infty}\kappa(t-\hat{t},s)\,I^{% \rm ext}(t-s)\,{\text{d}}s+u_{\rm rest}\,. (6.34)

6.4.3 Mapping the Integrate-and-Fire Model to the SRM

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 RR and capacity CC

τmduidt=-(ui-E0)-Rkwk+RIi(t)\tau_{m}\,{{\text{d}}u_{i}\over{\text{d}}t}=-(u_{i}-E_{0})-R\,\sum_{k}w_{k}+R% \,I_{i}(t) (6.35)

where τm=RC\tau_{m}=RC is the time constant, E0E_{0} the leak reversal potential, wkw_{k} are adaptation variables, and IiI_{i} is the input current to neuron ii. At each firing time

{ti(f)}{t|ui(t)=ϑ}.\{t_{i}^{(f)}\}\in\left\{t|u_{i}(t)=\vartheta\right\}\,. (6.36)

the voltage is reset to a value uru_{r}. At the same time, the adaptation variables are increased by an amount bkb_{k}

τkdwkdt=ak(ui-E0)-wk+τkbkt(f)δ(t-t(f))\tau_{k}\,\frac{{\text{d}}w_{k}}{{\text{d}}t}=a_{k}\,(u_{i}-E_{0})-w_{k}+\tau_% {k}\,b_{k}\sum_{t^{(f)}}\delta(t-t^{(f)}) (6.37)

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 Iiout=-qδ(t)I_{i}^{\rm out}=-q\,\delta(t) applied to the RCRC circuit. It removes a charge qq from the capacitor CC and lowers the potential by an amount Δu=-q/C\Delta u=-q/C. Thus, a reset of the membrane potential from a value of u=ϑu=\vartheta to a new value u=uru=u_{r} corresponds to an ‘output’ current pulse which removes a charge q=C(ϑ-ur)q=C\,(\vartheta-u_{r}). The reset takes place every time when the neuron fires. The total reset current is therefore

Iiout(t)=-C(ϑ-ur)fδ(t-ti(f)),I_{i}^{\rm out}(t)=-C\,({\vartheta-u_{r}})\,\sum_{f}\delta(t-t_{i}^{(f)})\,, (6.38)

where the sum runs over all firing times ti(f)t_{i}^{(f)}. We add the output current (6.38) on the right-hand side of (6.35),

τmduidt\displaystyle\tau_{m}\,{{\text{d}}u_{i}\over{\text{d}}t} =\displaystyle= -(ui-E0)-Rkwk+RIi(t)-RC(ϑ-ur)fδ(t-ti(f)),\displaystyle-(u_{i}-E_{0})-R\,\sum_{k}w_{k}+R\,I_{i}(t)-R\,C\,({\vartheta-u_{% r}})\,\sum_{f}\delta(t-t_{i}^{(f)})\,, (6.39)
τkdwkdt\displaystyle\tau_{k}\,\frac{{\text{d}}w_{k}}{{\text{d}}t} =\displaystyle= ak(ui-E0)-wk+τkbkt(f)δ(t-t(f)).\displaystyle a_{k}\,(u_{i}-E_{0})-w_{k}+\tau_{k}\,b_{k}\sum_{t^{(f)}}\delta(t% -t^{(f)}). (6.40)

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 KK adaptation variables, we have a total of K+1K+1 eigenvalues which we label as λ1,λ2\lambda_{1},\lambda_{2}\dots. The associated eigenvectors are 𝒆k\mbox{\boldmath\(e\)}_{k} with components (ek0,ek1,,ekK)T(e_{k0},e_{k1},\dots,e_{kK})^{T}. Third, we express the response to an impulse Δu=1\Delta u=1 in the voltage (no perturbation in the adaptation variables) in terms of the K+1K+1 eigenvectors: (1,0,0,,0)T=k=0Kβk𝒆k(1,0,0,...,0)^{T}=\sum_{k=0}^{K}\beta_{k}\mbox{\boldmath\(e\)}_{k}. Finally, we express the pulse caused by a reset of voltage and adaptation variables in terms of the eigenvectors (-ϑ+ur,b1,b2,,bK)T=k=0Kγk𝒆k(-\vartheta+u_{r},b_{1},b_{2},...,b_{K})^{T}=\sum_{k=0}^{K}\gamma_{k}\mbox{% \boldmath\(e\)}_{k}.

The response to the reset pulses yields the kernel η\eta while the response to voltage pulses yields the filter κ(s)\kappa(s) of the SRM

ui(t)\displaystyle u_{i}(t) =\displaystyle= fη(t-ti(f))+\displaystyle\sum_{f}\eta(t-t_{i}^{(f)})+ (6.41)

with kernels

η(s)\displaystyle\eta(s) =\displaystyle= k=0Kγk𝒆k0exp(λks)Θ(s),\displaystyle\sum_{k=0}^{K}\gamma_{k}\mbox{\boldmath\(e\)}_{k0}\,\exp(\lambda_% {k}\,s)\Theta(s)\,, (6.42)
κ(s)\displaystyle\kappa(s) =\displaystyle= k=0Kβk𝒆k0exp(λks).Θ(s)\displaystyle\sum_{k=0}^{K}\beta_{k}\mbox{\boldmath\(e\)}_{k0}\,\exp(\lambda_{% k}\,s)\,.\Theta(s) (6.43)

As usual, Θ(x)\Theta(x) denotes the Heaviside step function.

Fig. 6.13: SRM with a choice of η\eta leading to adaptation. A. The response of the neuron model to injection of a step current. B. The spike after-potential η\eta with adaptation time constant τw=100\tau_{w}=100 ms. A short (0.5 ms) period at +40 mV replaces the stereotypical shape of the action potential.

Example: Adaptation and Bursting

Let us first study a leaky integrate-and-fire model with a single slow adaptation variable τwτm\tau_{w}\gg\tau_{m} which is coupled to the voltage in the subthreshold regime (a>0a>0) and increased during spiking by an amount bb. In this case there are only two equations, one for the voltage and one for adaptation, so that the eigenvectors and eigenvalues can be calculated ‘by hand’. With a parameter δ=τm/τw1\delta=\tau_{m}/\tau_{w}\ll 1, the eigenvalues are λ1=-τw[1-aδ]\lambda_{1}=-\tau_{w}\,[1-a\,\delta] and λ2=-τwδ[1+a]\lambda_{2}=-\tau_{w}\,\delta\,[1+a], associated to eigenvectors 𝒆1=(1,aδ)T\mbox{\boldmath\(e\)}_{1}=(1,a\,\delta)^{T} and 𝒆2=(1,-1+δ+aδ)T\mbox{\boldmath\(e\)}_{2}=(1,-1+\delta+a\,\delta)^{T}. The resulting spike after-potential kernel η(s)\eta(s) is shown in Fig. 6.13B. Because of the slow time constant τwτm\tau_{w}\gg\tau_{m}, the kernel η\eta has a long hyperpolarizing tail. The neuron model responds to a step current with adaptation, because of accumulation of hyperpolarizing spike-after potentials over many spikes.

As a second example, we consider four adaptation currents with different time constants τ1<τ2<τ3<τ4\tau_{1}<\tau_{2}<\tau_{3}<\tau_{4}. We assume pure spike-triggered coupling (a=0a=0) so that the integration of the differential equations of wkw_{k} gives each an exponential current

wk(t)=fbkexp(-t-t(f)τk)Θ(t-t(f))w_{k}(t)=\sum_{f}b_{k}\,\exp(-{t-t^{(f)}\over\tau_{k}})\,\Theta(t-t^{(f)}) (6.44)

We choose the time constant of the first current to be very short and b1<0b_{1}<0 (inward current) so as to model the upswing of the action potential (a candidate current would be sodium). A second current (e.g. a fast potassium channel) with a slightly longer time constant is outgoing (b2>0b_{2}>0) and leads to the downswing and rapid reset of the membrane potential. The third current, with a time constant of tens of milliseconds is inward (b3<0b_{3}<0), while the slowest current is again hyperpolarizing (b4>0b_{4}>0). Integration of the voltage equation with all four currents generates the spike after-potential η\eta shown Fig. 6.14B. Because of the depolarizing spike after-potential induced by the inward current w3w_{3}, the neuron model responds to a step current of appropriate amplitude with bursts. The bursts end because of the accumulation of the hyperpolarizing effect of the slowest current.

Fig. 6.14: SRM with choice of η\eta leading to bursting. A. The refractory kernel η\eta of an integrate-and-fire model with four spike-triggered currents. B. The voltage response to a step current exhibits bursting. Adapted from Gerstner et al. (179).

6.4.4 Multi-compartment integrate-and-fire model as a SRM (*)

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 nn compartments. Membrane resistance, core resistance, and capacity of compartment μ\mu are denoted by RTμR_{\text{T}}^{\mu}, RLμR_{\text{L}}^{\mu}, and CμC^{\mu}, respectively. The longitudinal core resistance between compartment μ\mu and a neighboring compartment ν\nu is rμν=(RLμ+RLν)/2r^{\mu\nu}=(R_{\text{L}}^{\mu}+R_{\text{L}}^{\nu})/2; cf. Fig. 3.8. Compartment μ=1\mu=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 (2μn2\leq\mu\leq n) are passive.

Each compartment 1μn1\leq\mu\leq n of neuron ii may receive input Iiμ(t)I_{i}^{\mu}(t) from presynaptic neurons. As a result of spike generation, there is an additional reset current Ωi(t)\Omega_{i}(t) at the soma. The membrane potential ViμV_{i}^{\mu} of compartment μ\mu is given by

ddtViμ=1Ciμ[-ViμRT,iμ-νViμ-Viνriμν+Iiμ(t)-δμ 1Ωi(t)],{{\text{d}}\over{\text{d}}t}V_{i}^{\mu}=\frac{1}{C_{i}^{\mu}}\,\left[-{V_{i}^{% \mu}\over R_{\text{T},i}^{\mu}}-\sum_{\nu}{V_{i}^{\mu}-V_{i}^{\nu}\over r_{i}^% {\mu\nu}}+I_{i}^{\mu}(t)-\delta^{\mu\,1}\,\Omega_{i}(t)\right]\,, (6.45)

where the sum runs over all neighbors of compartment μ\mu. The Kronecker symbol δμν\delta^{\mu\nu} equals unity if the upper indices are equal; otherwise, it is zero. The subscript ii is the index of the neuron; the upper indices μ\mu or ν\nu refer to compartments. Below we will identify the somatic voltage Vi1V_{i}^{1} with the potential uiu_{i} 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 Giμν(s)G_{i}^{\mu\nu}(s) that describe the impact of an current pulse injected in compartment ν\nu on the membrane potential of compartment μ\mu. The solution is of the form

Viμ(t)=ν1Ciν0Giμν(s)[Iiν(t-s)-δν1Ωi(t-s)]ds.V_{i}^{\mu}(t)=\sum_{\nu}\frac{1}{C_{i}^{\nu}}\,\int_{0}^{\infty}G_{i}^{\mu\nu% }(s)\left[I_{i}^{\nu}(t-s)-\delta^{\nu 1}\,\Omega_{i}(t-s)\right]\,{\text{d}}s\,. (6.46)

Explicit expressions for the Green’s function Giμν(s)G_{i}^{\mu\nu}(s) for arbitrary geometry have been derived by Abbott et al. (1) and Bressloff and Taylor (66).

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 tj(f)t_{j}^{(f)} of a presynaptic neuron jj evokes, for t>tj(f)t>t_{j}^{(f)}, a synaptic current pulse α(t-tj(f))\alpha(t-t_{j}^{(f)}) into the postsynaptic neuron ii. The actual amplitude of the current pulse depends on the strength WijW_{ij} of the synapse that connects neuron jj to neuron ii. The total input to compartment μ\mu of neuron ii is thus

Iiμ(t)=jΓiμWijfα(t-tj(f)).I_{i}^{\mu}(t)=\sum_{j\in\Gamma_{i}^{\mu}}W_{ij}\,\sum_{f}\alpha(t-t_{j}^{(f)}% )\,. (6.47)

Here, Γiμ\Gamma_{i}^{\mu} denotes the set of all neurons that have a synapse with compartment μ\mu of neuron ii. The firing times of neuron jj are denoted by tj(f)t_{j}^{(f)}.

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, ϑ\vartheta. After each spike the somatic membrane potential is reset to Vi1=ur<ϑV_{i}^{1}=u_{r}<\vartheta. This is equivalent to a current pulse

γi(s)=Ci1(ϑ-ur)δ(s),\gamma_{i}(s)=C_{i}^{1}\,(\vartheta-u_{r})\,\delta(s)\,, (6.48)

so that the overall current due to the firing of action potentials at the soma of neuron ii amounts to

Ωi(t)=fγi(t-ti(f)).\Omega_{i}(t)=\sum_{f}\gamma_{i}(t-t_{i}^{(f)})\,. (6.49)

We will refer to equations (6.46)–(6.49) together with the threshold criterion for generating spikes as the multi-compartment integrate-and-fire model.

Using the above specializations for the synaptic input current and the somatic reset current the membrane potential (6.46) of compartment μ\mu in neuron ii can be rewritten as

Viμ(t)=fηiμ(t-ti(f))+νjΓiνWijfϵiμν(t-tj(f)).V_{i}^{\mu}(t)=\sum_{f}\eta_{i}^{\mu}(t-t_{i}^{(f)})+\sum_{\nu}\sum_{j\in% \Gamma_{i}^{\nu}}W_{ij}\sum_{f}\epsilon_{i}^{\mu\nu}(t-t_{j}^{(f)}). (6.50)


ϵiμν(s)\displaystyle\epsilon_{i}^{\mu\nu}(s) =1Ciν0Giμν(s)α(s-s)ds,\displaystyle=\frac{1}{C_{i}^{\nu}}\,\int_{0}^{\infty}G_{i}^{\mu\nu}(s^{\prime% })\,\alpha(s-s^{\prime})\,{\text{d}}s^{\prime}\,, (6.51)
ηiμ(s)\displaystyle\eta_{i}^{\mu}(s) =1Ci10Giμ1(s)γi(s-s)ds.\displaystyle=\frac{1}{C_{i}^{1}}\,\int_{0}^{\infty}G_{i}^{\mu 1}(s^{\prime})% \,\gamma_{i}(s-s^{\prime})\,{\text{d}}s^{\prime}. (6.52)

The kernel ϵiμν(s)\epsilon_{i}^{\mu\nu}(s) describes the effect of a presynaptic action potential arriving at compartment ν\nu on the membrane potential of compartment μ\mu. Similarly, ηiμ(s)\eta_{i}^{\mu}(s) describes the response of compartment μ\mu to an action potential generated at the soma.

The triggering of action potentials depends on the somatic membrane potential only. We define ui=Vi1u_{i}=V_{i}^{1}, ηi(s)=ηi1(s)\eta_{i}(s)=\eta_{i}^{1}(s) and, for jΓiνj\in\Gamma_{i}^{\nu}, we set ϵij=ϵi1ν\epsilon_{ij}=\epsilon_{i}^{1\nu}. This yields the equation of the SRM

ui(t)=fηi(t-ti(f))+jWijfϵij(t-tj(f)).u_{i}(t)=\sum_{f}\eta_{i}(t-t_{i}^{(f)})+\sum_{j}W_{ij}\sum_{f}\epsilon_{ij}(t% -t_{j}^{(f)}). (6.53)
Fig. 6.15: Two-compartment integrate-and-fire model. A. Response kernel η0(s)\eta_{0}(s) of a neuron with two compartments and a fire-and-reset threshold dynamics. The response kernel is a double exponential with time constants τ12=2\tau_{12}=2 ms and τ0=10\tau_{0}=10~{}ms. The spike at s=0s=0 is indicated by a vertical arrow. B. Response kernel ϵ0(s)\epsilon_{0}(s) for excitatory synaptic input at the dendritic compartment with a synaptic time constant τs=1\tau_{s}=1~{}ms. The response kernel is a superposition of three exponentials and exhibits the typical time course of an excitatory postsynaptic potential.

Example: Two-compartment integrate-and-fire model

We illustrate the methodology by mapping a simple model with two compartments and a reset mechanism at the soma (446) to the Spike Response Model. The two compartments are characterized by a somatic capacitance C1C^{1} and a dendritic capacitance C2=aC1C^{2}=a\,C^{1}. The membrane time constant is τ0=R1C1=R2C2\tau_{0}=R^{1}\,C^{1}=R^{2}\,C^{2} and the longitudinal time constant τ12=r12C1C2/(C1+C2)\tau_{12}=r^{12}\,C^{1}\,C^{2}/(C^{1}+C^{2}). The neuron fires, if V1(t)=ϑV^{1}(t)=\vartheta. After each firing the somatic potential is reset to uru_{r}. This is equivalent to a current pulse

γ(s)=qδ(s),\gamma(s)=q\,\delta(s)\,, (6.54)

where q=C1[ϑ-ur]q=C^{1}\,[\vartheta-u_{r}] is the charge lost during the spike. The dendrite receives spike trains from other neurons jj and we assume that each spike evokes a current pulse with time course

α(s)=1τsexp(-sτs)Θ(s).\alpha(s)={1\over\tau_{s}}\exp\left(-\frac{s}{\tau_{s}}\right)\Theta(s)\,. (6.55)

For the two-compartment model it is straightforward to integrate the equations and derive the Green’s function. With the Green’s function we can calculate the response kernels η0(s)=ηi(1)\eta_{0}(s)=\eta_{i}^{(1)} and ϵ0(s)=ϵi12\epsilon_{0}(s)=\epsilon_{i}^{12} as defined in Eqs. (6.51) and (6.52). We find

η0(s)\displaystyle\eta_{0}(s) =\displaystyle= -ϑ-ur(1+a)exp(-sτ0)[1+aexp(-sτ12)],\displaystyle-{\vartheta-u_{r}\over(1+a)}\,\exp\left(-{s\over\tau_{0}}\right)% \,\left[1+a\,\exp\left(-{s\over\tau_{12}}\right)\right]\,, (6.56)
ϵ0(s)\displaystyle\epsilon_{0}(s) =\displaystyle= 1(1+a)exp(-sτ0)[1-e-δ1sτsδ1-exp(-sτ12)1-e-δ2sτsδ2],\displaystyle{1\over(1+a)}\exp\left(-{s\over\tau_{0}}\right)\,\left[{1-e^{-{% \delta_{1}s}}\over\tau_{s}\,\delta_{1}}-\exp\left(-{s\over\tau_{12}}\right)\,{% 1-e^{-{\delta_{2}s}}\over\tau_{s}\,\delta_{2}}\right]\,,

with δ1=τs-1-τ0-1\delta_{1}=\tau_{s}^{-1}-\tau_{0}^{-1} and δ2=τs-1-τ0-1-τ12-1\delta_{2}=\tau_{s}^{-1}-\tau_{0}^{-1}-\tau_{12}^{-1}. Figure 6.15 shows the two response kernels with parameters τ0=10\tau_{0}=10~{}ms, τ12=2\tau_{12}=2 ms, and a=10a=10. The synaptic time constant is τs=1\tau_{s}=1 ms. The kernel ϵ0(s)\epsilon_{0}(s) describes the voltage response of the soma to an input at the dendrite. It shows the typical time course of an excitatory or inhibitory postsynaptic potential. The time course of the kernel η0(s)\eta_{0}(s) is a double exponential and reflects the dynamics of the reset in a two-compartment model.