Subsections

• 4.4.1 Definition of the Model
• 4.4.2 Relation to the Model SRM₀
  • 4.4.2.1 Example: Two-compartment integrate-and-fire model
  
  
• 4.4.3 Relation to the Full Spike Response Model (*)

4.4 Multi-compartment integrate-and-fire model

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.

4.4.1 Definition of the 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 $\mu$ are denoted by R_T^$\mu$, R_L^$\mu$, and C^$\mu$, respectively. The longitudinal core resistance between compartment $\mu$ and a neighboring compartment $\nu$ is r^$\mu$$\nu$ = (R_L^$\mu$ + R_L^$\nu$)/2; cf. Fig. . Compartment $\mu$ = 1 represents the soma and is equipped with a simple mechanism for spike generation, i.e., with a threshold criterion as in the standard integrate-and-fire model. The remaining dendritic compartments ( 2$\le$$\mu$$\le$n) are passive.

Each compartment 1$\le$$\mu$$\le$n of neuron i may receive input I_i^$\mu$(t) from presynaptic neurons. As a result of spike generation, there is an additional reset current $\Omega_{i}^{}$(t) at the soma. The membrane potential V_i^$\mu$ of compartment $\mu$ is given by

$${{\text{d}}\over {\text{d}}t} \mu {\frac{{1}}{{C_i^\mu}}} \left[\vphantom{ - {V_i^\mu \over R_{\text{T},i}^\mu} + \sum_\nu ... ...nu \over r_i^{\mu\nu}} + I_i^\mu (t) - \delta^{\mu\,1} \, \Omega_i (t) }\right. {V_i^\mu \over R_{\text{T},i}^\mu} \sum_{\nu}^{} {V_i^\mu - V_i^\nu \over r_i^{\mu\nu}} \mu \delta^{{\mu\,1}}_{} \Omega_{i}^{} \left.\vphantom{ - {V_i^\mu \over R_{\text{T},i}^\mu} + \sum_\nu ... ...nu \over r_i^{\mu\nu}} + I_i^\mu (t) - \delta^{\mu\,1} \, \Omega_i (t) }\right]$$ (4.81)

where the sum runs over all neighbors of compartment $\mu$. The Kronecker symbol $\delta^{{\mu\nu}}_{}$ equals unity if the upper indices are equal; otherwise, it is zero. The subscript i is the index of the neuron; the upper indices $\mu$ or $\nu$ refer to compartments. Below we will identify the somatic voltage V_i¹ with the potential u_i of the Spike Response Model.

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^$\mu$$\nu$(s) that describe the impact of an current pulse injected in compartment $\nu$ on the membrane potential of compartment $\mu$. The solution is of the form

$$\mu \sum_{\nu}^{} {\frac{{1}}{{C_i^\nu}}} \int_{0}^{\infty} \mu \nu \left[\vphantom{ I_i^\nu (t-s) - \delta^{\nu 1} \, \Omega_i (t-s) }\right. \nu \delta^{{\nu 1}}_{} \Omega_{i}^{} \left.\vphantom{ I_i^\nu (t-s) - \delta^{\nu 1} \, \Omega_i (t-s) }\right]$$ (4.82)

Explicit expressions for the Green's function G_i^$\mu$$\nu$(s) for arbitrary geometry have been derived by Abbott et al. (1991) and Bressloff and Taylor (1994).

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 $\alpha$(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 $\mu$ of neuron i is thus

$$\mu \sum_{{j \in \Gamma_i^\mu}}^{} \sum_{f}^{} \alpha$$ (4.83)

Here, $\Gamma_{i}^{\mu}$ denotes the set of all neurons that have a synapse with compartment $\mu$ of neuron i. The firing times of neuron j are denoted by 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, $\vartheta$. After each spike the somatic membrane potential is reset to V_i¹ = u_r < $\vartheta$. This is equivalent to a current pulse

$$\gamma_{i}^{} \vartheta \delta$$ (4.84)

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

$$\Omega_{i}^{} \sum_{f}^{} \gamma_{i}^{}$$ (4.85)

We will refer to equations (4.82)-(4.85) together with the threshold criterion for generating spikes as the multi-compartment integrate-and-fire model.

4.4.2 Relation to the Model SRM₀

Using the above specializations for the synaptic input current and the somatic reset current the membrane potential (4.82) of compartment $\mu$ in neuron i can be rewritten as

$$\mu \sum_{f}^{} \eta_{i}^{\mu} \sum_{\nu}^{} \sum_{{j \in \Gamma_i^\nu}}^{} \sum_{f}^{} \epsilon_{i}^{{\mu\nu}}$$ (4.86)

with

$$\epsilon_{i}^{{\mu\nu}} {\frac{{1}}{{C_i^\nu}}} \int_{0}^{\infty} \mu \nu \alpha$$ (4.87)

$$\eta_{i}^{\mu} {\frac{{1}}{{C_i^1}}} \int_{0}^{\infty} \mu \gamma_{i}^{}$$ (4.88)

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

The triggering of action potentials depends on the somatic membrane potential only. We define u_i = V_i¹, $\eta_{i}^{}$(s) = $\eta_{i}^{1}$(s) and, for j $\in$ $\Gamma_{i}^{\nu}$, we set $\epsilon_{{ij}}^{}$ = $\epsilon_{i}^{{1 \nu}}$. This yields

$$\sum_{f}^{} \eta_{i}^{} \sum_{{j}}^{} \sum_{f}^{} \epsilon_{{ij}}^{}$$ (4.89)

As in (4.54), we use a short-term memory approximation and truncate the sum over the $\eta_{i}^{}$-terms. The result is

$$\eta_{i}^{} \hat{{t_i}} \sum_{j}^{} \sum_{f}^{} \epsilon_{{ij}}^{}$$ (4.90)

where $\hat{{t_i}}$ is the last firing time of neuron i. Thus, the multi-compartment model has been reduced to the single-variable model of Eq. (4.42). The approximation is good, if the typical inter-spike interval is long compared to the neuronal time constants.

4.4.2.1 Example: Two-compartment integrate-and-fire model

Figure 4.24: Two-compartment integrate-and-fire model. A. Response kernel $\eta_{0}^{}$(s) of a neuron with two compartments and a fire-and-reset threshold dynamics. The response kernel is a double exponential with time constants $\tau_{{12}}^{}$ = 2ms and $\tau_{0}^{}$ = 10 ms. The spike at s = 0 is indicated by a vertical arrow. B. Response kernel $\epsilon_{0}^{}$(s) for excitatory synaptic input at the dendritic compartment with a synaptic time constant $\tau_{s}^{}$ = 1 ms. The response kernel is a superposition of three exponentials and exhibits the typical time course of an excitatory postsynaptic potential.

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 C¹ and a dendritic capacitance C² = a C¹. The membrane time constant is $\tau_{0}^{}$ = R¹ C¹ = R² C² and the longitudinal time constant $\tau_{{12}}^{}$ = r¹² C¹ C²/(C¹ + C²). The neuron fires, if V¹(t) = $\vartheta$. After each firing the somatic potential is reset to u_r. This is equivalent to a current pulse

$$\gamma \delta$$ (4.91)

where q = C¹ [$\vartheta$ - u_r] is the charge lost during the spike. The dendrite receives spike trains from other neurons j and we assume that each spike evokes a current pulse with time course

$$\alpha {1\over \tau_s} \left(\vphantom{- \frac{s}{\tau_s} }\right. {\frac{{s}}{{\tau_s}}} \left.\vphantom{- \frac{s}{\tau_s} }\right) \Theta$$ (4.92)

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 $\eta_{0}^{}$(s) = $\eta_{i}^{{(1)}}$ and $\epsilon_{0}^{}$(s) = $\epsilon_{i}^{{12}}$ as defined in Eqs. (4.87) and (4.88); cf. Tuckwell (1988), Bressloff and Taylor (1994). We find

$$\eta_{0}^{} {\vartheta - u_r \over (1+a)} \left(\vphantom{-{s\over \tau_0} }\right. {s\over \tau_0} \left.\vphantom{-{s\over \tau_0} }\right) \left[\vphantom{1 + a \, \exp\left(-{s\over \tau_{12}} \right) }\right. \left(\vphantom{-{s\over \tau_{12}} }\right. {s\over \tau_{12}} \left.\vphantom{-{s\over \tau_{12}} }\right) \left.\vphantom{1 + a \, \exp\left(-{s\over \tau_{12}} \right) }\right]$$ = (4.93)

$$\epsilon_{0}^{} {1 \over (1+a)} \left(\vphantom{-{s\over \tau_0} }\right. {s\over \tau_0} \left.\vphantom{-{s\over \tau_0} }\right) \left[\vphantom{ {1 - e^{-{\delta_1 s}} \over \tau_s \, \delta_1}... ...er \tau_{12}}\right)\, {1 - e^{-{\delta_2 s}} \over \tau_s\, \delta_2} }\right. {1 - e^{-{\delta_1 s}} \over \tau_s \, \delta_1} \left(\vphantom{-{s\over \tau_{12}} }\right. {s\over \tau_{12}} \left.\vphantom{-{s\over \tau_{12}} }\right) {1 - e^{-{\delta_2 s}} \over \tau_s\, \delta_2} \left.\vphantom{ {1 - e^{-{\delta_1 s}} \over \tau_s \, \delta_1}... ...er \tau_{12}}\right)\, {1 - e^{-{\delta_2 s}} \over \tau_s\, \delta_2} }\right]$$ =

with $\delta_{1}^{}$ = $\tau_{s}^{{-1}}$ - $\tau_{0}^{{-1}}$ and $\delta_{2}^{}$ = $\tau_{s}^{{-1}}$ - $\tau_{0}^{{-1}}$ - $\tau_{{12}}^{{-1}}$. Figure 4.24 shows the two response kernels with parameters $\tau_{0}^{}$ = 10 ms, $\tau_{{12}}^{}$ = 2 ms, and a = 10. The synaptic time constant is $\tau_{s}^{}$ = 1 ms. The kernel $\epsilon_{0}^{}$(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 $\eta_{0}^{}$(s) is a double exponential and reflects the dynamics of the reset in a two-compartment model.

4.4.3 Relation to the Full Spike Response Model (*)

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 $\epsilon$ 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 $\hat{{t}}_{i}^{}$. Unfortunately, the set of available initial conditions at $\hat{{t}}_{i}^{}$ is incomplete because only the somatic membrane potential equals u_r immediately after t = $\hat{{t_i}}$. For the membrane potential of the remaining compartments we have to use initial conditions at t = - $\infty$, 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 = $\hat{{t}}_{i}^{}$ into two parts,

$$\sum_{{\nu}}^{} {\frac{{1}}{{C_i^\nu}}} \int_{{-\infty}}^{{\hat{t}_i+0}} \nu \left[\vphantom{ I_i^\nu(s) - \delta^{1\nu} \, \Omega_i(s) }\right. \nu \delta^{{1\nu}}_{} \Omega_{i}^{} \left.\vphantom{ I_i^\nu(s) - \delta^{1\nu} \, \Omega_i(s) }\right]$$ V_i¹(t)

$$\sum_{{\nu}}^{} {\frac{{1}}{{C_i^\nu}}} \int_{{\hat{t}_i+0}}^{{t}} \nu \nu$$ (4.94)

The limits of the integration have been chosen to be at $\hat{{t}}_{i}^{}$ + 0 in order to ensure that the Dirac $\delta$-pulse for the reset of the membrane potential is included in the first term.

With G_i^1$\nu$(t - s) = $\sum_{\mu}^{}$G_i^1$\mu$(t - $\hat{{t}}$) G_i^$\mu$$\nu$($\hat{{t}}$ - s), which is a general property of Green's functions, we obtain

$$\sum_{{\mu}}^{} \mu \hat{{t}}_{i}^{} \sum_{{\nu}}^{} {\frac{{1}}{{C_i^\nu}}} \int_{{-\infty}}^{{\hat{t}_i+0}} \mu \nu \hat{{t}}_{i}^{} \left[\vphantom{ I_i^\nu(s) - \delta^{1\nu} \, \Omega_i(s) }\right. \nu \delta^{{1\nu}}_{} \Omega_{i}^{} \left.\vphantom{ I_i^\nu(s) - \delta^{1\nu} \, \Omega_i(s) }\right]$$ V_i¹(t)

$$\sum_{{\nu}}^{} {\frac{{1}}{{C_i^\nu}}} \int_{{\hat{t}_i+0}}^{{t}} \nu \nu$$ (4.95)

With the known initial condition at the soma,

$$\hat{{t}}_{i}^{} \sum_{{\nu}}^{} {\frac{1}{{C_i^\nu}}} \int_{{-\infty}}^{{\hat{t}_i+0}} \nu \hat{{t}}_{i}^{} \left[\vphantom{ I_i^\nu(s) - \delta^{1\nu} \, \Omega_i(s) }\right. \nu \delta^{{1\nu}}_{} \Omega_{i}^{} \left.\vphantom{ I_i^\nu(s) - \delta^{1\nu} \, \Omega_i(s) }\right]$$ (4.96)

we find

$$\hat{{t}}_{i}^{}$$ V_i¹(t)

$$\sum_{{\mu \ge 2}}^{} \sum_{{\nu}}^{} {\frac{{1}}{{C_i^\nu}}} \mu \hat{{t}}_{i}^{} \int_{{-\infty}}^{{\hat{t}_i+0}} \mu \nu \hat{{t}}_{i}^{} \left[\vphantom{ I_i^\nu(s) - \delta^{1\nu} \, \Omega_i(s) }\right. \nu \delta^{{1\nu}}_{} \Omega_{i}^{} \left.\vphantom{ I_i^\nu(s) - \delta^{1\nu} \, \Omega_i(s) }\right]$$

$$\sum_{{\nu}}^{} {\frac{{1}}{{C_i^\nu}}} \int_{{\hat{t}_i+0}}^{{t}} \nu \nu$$ (4.97)

The voltage reset at the soma is described by $\Omega_{i}^{}$(t) = C_i¹ ($\vartheta$ - u_r)$\sum_{f}^{}$$\delta$(t - t_i^(f)); cf. Eqs. (4.84) and (4.85). After shifting the terms with $\Omega$ to the end and substituting its definition, we obtain

$$\hat{{t}}_{i}^{}$$ V_i¹(t)

$$\sum_{{\mu \ge 2}}^{} \sum_{{\nu}}^{} {\frac{{1}}{{C_i^\nu}}} \mu \hat{{t}}_{i}^{} \int_{{-\infty}}^{{\hat{t}_i+0}} \mu \nu \hat{{t}}_{i}^{} \nu$$

$$\sum_{{\nu}}^{} {\frac{{1}}{{C_i^\nu}}} \int_{{\hat{t}_i+0}}^{{t}} \nu \nu$$

$$\vartheta \sum_{{\mu \ge 2}}^{} \sum_{f}^{} \mu \hat{{t}}_{i}^{} \mu \hat{{t}}_{i}^{}$$ (4.98)

If we introduce

$$\tilde{{G}}_{i}^{{1\nu}} \begin{cases} \frac{1}{C_i^\nu} \, G_i^{1\nu}(s) & r > s \\ ... ...u \ge 2} G_i^{1\mu}(r) \, G_i^{\mu\nu}(s-r) & r < s \end{cases}$$ (4.99)

we can collect the integrals in Eq. (4.98) and obtain

$$\hat{{t}}_{i}^{}$$ V_i¹(t)

$$\sum_{{\nu}}^{} \int_{{-\infty}}^{{t}} \tilde{{G}}_{i}^{{1\nu}} \hat{{t}}_{i}^{} \nu$$

$$\vartheta \sum_{{\mu \ge 2}}^{} \sum_{f}^{} \mu \hat{{t}}_{i}^{} \mu \hat{{t}}_{i}^{}$$ (4.100)

This expression has a clear interpretation. The first term describes the relaxation of the somatic membrane potential in the absence of further input. The second term accounts for external input to any of the compartments integrated up to time t. Finally, the last term reflects an indirect influence of previous spikes on the somatic membrane potential via other compartments that are not reset during an action potential. In fact, the sum over the firing times in the last term stops at the last but one action potential since G_i^$\mu$1($\hat{{t}}_{i}^{}$ - t_i^(f)), $\mu$ > 1, is zero if $\hat{{t}}_{i}^{}$ = t_i^(f).

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

$$\epsilon_{i}^{\nu} \int_{0}^{\infty} \tilde{{G}}_{i}^{{1\nu}} \alpha$$ (4.101)

and

$$\eta_{i}^{}$$ (4.102)

Here, $\alpha$(s) describes the form of an elementary postsynaptic current; cf. Eq. (4.83). With these definitions the somatic membrane potential u_i(t) $\equiv$ V_i¹(t) of neuron i is

$$\eta_{i}^{} \hat{{t}}_{i}^{} \sum_{{\nu}}^{} \sum_{{j \in \Gamma_i^\nu}}^{} \sum_{f}^{} \epsilon_{i}^{\nu} \hat{{t}}_{i}^{}$$ (4.103)

which is the equation of the Spike Response Model.