On dependency properties of the ISIs generated by a two-compartmental neuronal model

Abstract

One-dimensional leaky integrate and fire neuronal models describe interspike intervals (ISIs) of a neuron as a renewal process and disregarding the neuron geometry. Many multi-compartment models account for the geometrical features of the neuron but are too complex for their mathematical tractability. Leaky integrate and fire two-compartment models seem a good compromise between mathematical tractability and an improved realism. They indeed allow to relax the renewal hypothesis, typical of one-dimensional models, without introducing too strong mathematical difficulties. Here, we pursue the analysis of the two-compartment model studied by Lansky and Rodriguez (Phys D 132:267–286, 1999), aiming of introducing some specific mathematical results used together with simulation techniques. With the aid of these methods, we investigate dependency properties of ISIs for different values of the model parameters. We show that an increase of the input increases the strength of the dependence between successive ISIs.

Appendix

1.1 A.1 Proof of Eq. (13)

When $X_{2}^{B}(\tau )\approx X_2(\tau )$, $\tau \in (t_{i-1},t_{i})$, with $t_{i-1}$ and $t_i$ firing times, Eq. (12) becomes

$$\begin{aligned} X_2(t_i)-X_2(t_{i-1})&= -(\alpha +\alpha _r)\int \limits _{t_{i-1}}^{t_i}X_2(t)\mathrm{d}t\nonumber \\&+\alpha _r\int \limits _{t_{i-1}}^{t_i}X_1(t)\mathrm{d}t. \end{aligned}$$

(24)

Since $X_2(t_i)=S$ and $X_2(t_{i-1}^+)=0$, Eq. (24) can be rewritten as

$$\begin{aligned} S=-(\alpha +\alpha _r)\int \limits _{t_{i-1}}^{t_i}X_2(t)\mathrm{d}t+\alpha _r\int \limits _{t_{i-1}}^{t_i}X_1(t)\mathrm{d}t. \end{aligned}$$

(25)

Taking the expectation of each member of (25) and applying Fubini’s theorem (cf.Folland 1999), we get

$$\begin{aligned} S=-(\alpha +\alpha _r)\left[\int \limits _{t_{i-1}}^{t_i}m_2(t)\mathrm{d}t\right]+\alpha _r\left[\int \limits _{t_{i-1}}^{t_i}m_1(t)\mathrm{d}t\right]. \end{aligned}$$

(26)

Now, considering the expressions (2a) and (2b) with initial condition $m_2(t_{i-1})=0$ and $m_1(t_{i-1})=M_{i-1}$, we have

$$\begin{aligned} m_1(t)=&m_1(\infty )+ \frac{1}{2}\left(M_{i-1}-\frac{\mu }{\alpha }\right) \mathrm{e}^{-\alpha t}\nonumber \\&+\frac{1}{2}\left(M_{i-1}-\frac{\mu }{\alpha +2\alpha _r}\right) \mathrm{e}^{-(\alpha +2 \alpha _r) t}\end{aligned}$$

(27)

$$\begin{aligned} m_2(t)=&m_2(\infty )+ \frac{1}{2}\left(M_{i-1}-\frac{\mu }{\alpha }\right) \mathrm{e}^{-\alpha t}\nonumber \\&+\frac{1}{2}\left(\frac{\mu }{\alpha +2\alpha _r}-M_{i-1}\right) \mathrm{e}^{-(\alpha +2\alpha _r) t} \end{aligned}$$

(28)

Finally, replacing (27) and (28) into (26), we get the following equation for the $i$th ISI, $i\ge 2$

$$\begin{aligned} 2(S-m_2(\infty ))&= \left\{ M_{i-1}-\frac{\mu }{\alpha }\right\} \mathrm{e}^{-\alpha T_i}\nonumber \\&+\left\{ \frac{\mu }{\alpha +2\alpha _r}-M_{i-1}\right\} \mathrm{e}^{-(\alpha +2\alpha _r)T_i}. \end{aligned}$$

(29)

1.2 A.2 The bivariate copula

Copulas are mathematical objects increasingly used to describe the joint behaviour of random vectors. We introduce here only the material necessary for this paper while we refer to Nelsen (1999) for a detailed introduction.

A bivariate copula is the joint cumulative distribution function of a bivariate random vector $(U,V)$ on the unit square $[0,1]\times [0,1]$ with uniform marginals:

$$\begin{aligned} C(u,v)=\mathbb P (U\le u, V\le v). \end{aligned}$$

If $F_1(x_1)$ and $F_2(x_2)$ are the marginal distribution functions of the random variables $X_1$ and $X_2$, then

$$\begin{aligned} C(F_1(x_1),F_2(x_2))=F(x_1,x_2) \end{aligned}$$

(30)

defines a bivariate distribution function with marginals $F_1(x_1)$ and $F_2(x_2)$. Sklar (1959) established also that the converse is true. Indeed he proved that any bivariate distribution function $F$ can be written in the form (30). Moreover, if the marginal distributions are continuous, the copula representation (30) is unique.

Copulas separate the study of dependency properties from the study of marginals. On the contrary, this two features are mixed in the joint distribution. Moreover, copulas are invariant under increasing and continuous transformations, i.e. they are scale free.

There exist different types of copulas, corresponding to different dependency structures. One example is the Gaussian copula associated to a multivariate normal distribution. It is constructed by projecting a bivariate normal distribution on the unit square $[0,1]^2$. For a given $2\times 2$ correlation matrix $\Sigma$, the Gaussian copula is

$$\begin{aligned} C_\Sigma (u,v)=\phi _\Sigma \left(\phi ^{-1}(u),\phi ^{-1}(v)\right). \end{aligned}$$

Here $\phi ^{-1}$ denotes the inverse cumulative distribution function of a standard normal and $\phi _\Sigma$ is the joint cumulative distribution function of a bivariate normal distribution with mean vector zero and covariance matrix equal to $\Sigma$.

Note that if in (30) one uses a Gaussian copula and non Gaussian marginal distributions, the joint distribution is not a bivariate normal distribution.