A two-state neuronal model with alternating exponential excitation

1.   The model of neural activity with alternating states

A neuron is surrounded by a membrane with selective conductivity depending on its current state. The membrane potential $V = V(t)$ undergoes a sudden change, which is called the "spike" potential. The spike (impulse) is generated in response to an external influence and only when it exceeds a certain threshold. Since most cells of central nervous system are characterised by spontaneously emitted train of impulses, stochastic modelling is of primary interest in the field. It is assumed that the excitatory stimuli that occur in random time are random and depend on the current state of the neuron. Each stimulus is followed by a refractory period, which corresponds to an exponential decay of the potential. In view of this, the first-passage-time problem for the underlying stochastic processes is important for the description of the neuronal firing.

A large number of models of single neurons were developed: from simple threshold models to biologically plausible "portrait" models. Best results were achieved when complicated experimental features were combined with a rather simple mathematical model, see, for example, the mathematical model of the Nobel Prize winner Hodgkin's of the squid giant axon by ^[1]. The first simple threshold neuron model is considered to be the model proposed by Louis Lapicque in 1907 ^[2] which is usually called the "leaky integrator" or the "forgetful integrate-and-fire" model. In modern terms, the simplest version of this model with an external forcing term described by a Brownian motion $W = W(t)$ yields the following stochastic differential equation

A spike is generated, once the process $V(t)$ hits the firing threshold $H$.

Since excitatory stimuli are intermittent, synaptic input should be modelled by means of point processes. This approach was proposed by R. Stein ^[3], see also ^[4]. In ^[5], Stein's model is briefly presented in a rigorous manner. See also recent paper ^[6] on this subject.

Stein's model presumes a time evolution of the potential described by a stochastic equation based on two independent Poisson processes $N_+$ and $N_-$

where $a_\pm$ denote the amplitudes of excitatory/inhibitory currents.

An excitatory model with an exponentially decaying membrane potential and a non-homogeneous Poisson process driving the consecutive neuronal stimuli is studied by ^[7]. Methods for assessing how well this model describes neural spikes are based on the time-rescaling technique, ^[8].

The detailed review of the existing models can be found in ^[9].

The model proposed in this paper suggests that neurons take one of two states, alternating at each stimulus time. Similar ideas are widely used in neural modelling. For example, in the recent monograph ^[10], the three-phase Stein model was presented: the standard Stein model is supplemented by an additional 0-phase, which starts at the end of the refractory period and lasts until depolarisation occurs. In ^[11], a two-phase model is studied, based on neuronal oscillations interrupted by stochastic behaviour. The authors claim that this can be explained by a bistability in the ensemble dynamics of coupled integrate and fire neurons. See also the paper ^[12], where some practical observations are presented that can serve as the basis for such an approach.

To describe the model, consider the right-continuous process ${\varepsilon} = {\varepsilon}(t),$ $t\geq0,$ which has the state space $\{0, \; 1\},$ with independent consecutive (random) holding times $\{T_n\}_{n\geq1}$. Let $N = N(t)$ be the process counting the switching of ${\varepsilon}$ till time $t$,

To construct a model of neural activity, we will use a well-studied class of jump-telegraph stochastic processes, see, for example, the review in ^[13] and ^[14]. Recent paper ^[15] intensively developed methods for studying the distributions of first passage times for such processes.

The jump-telegraph process $X = X(t), \; t\geq0,$ with additive jumps, is defined by the stochastic equation

which by integration yields

Here velocities $c_0, \; c_1$ are two real constants, $Y_n, \; n\geq1$ are independent random variables independent of ${\varepsilon}$, corresponding to jumps which accompany each velocity switching. By $X_i(t), \; t\geq0,$ we denote the solution of (1.1), provided with the additional initial condition: ${\varepsilon}(0) = i, \; i\in\{0, 1\}.$

We propose a neural potential model that includes state-dependent decay rates of membrane potential, along with multiplicative state-dependent stimuli. The model suggests a change in state at the each stimulus time.

This model presumes the nerve cell depolarisation $V = V(t),$ $t\geq0,$ to be determined by the stochastic exponential of $X,$ see (1.1)-(1.2). In other words, process $V = V(t)$ is the solution of the stochastic equation

The counting process $N = N(t)$ corresponds to the number of excitatory stimuli received by the neuron till time $t,$ $Y_n$ is the voltage level at the stimulus time, and the negative constants $c_0, \; c_1$ correspond to the rates of exponential decay of the membrane potential being in the state $0$ and $1$ respectively.

The solution $V = V(t)$ of (1.3) is given by stochastic exponential of $X$,

Let the holding times $T_n, \; n\geq1,$ have alternating distributions $\pi_0$ and $\pi_1,$ that is $\pi_i({{\rm{d}}} t) = {\mathbb P}\{T_1\in {{\rm{d}}} t\; |\; {\varepsilon}(0) = i\}, \; i\in\{0, 1\},$ and

Jumps $Y_n$ have the distributions $g_0$ and $g_1,$ alternately, together with alternating distributions $\pi_0$ and $\pi_1$ of holding times.

For each $t, \; t > 0,$ the distribution of $V(t)$ can be expressed by means of the given distributions $\pi_0, \pi_1, \; g_0, g_1,$ see e. g. ^[15].

The main interest of the neural modelling lies in the properties of the first passage time of the depolarisation process $V = V(t)$ through the firing threshold $H, \; H > 0.$ We are interested to study the distribution of of the stopping time ${\mathscr T}_x, \; x = \log H/V_0,$ of the jump-telegraph process,

For a continuous version of such processes, $Y_n = 0\; \forall n,$ this mathematical problem is well known and studied, see e.g. ^[16,17,18]. See also the recent paper ^[19], where arbitrary sequences of velocities and jumps intensities were applied. The number of level-crossings for the telegraph process has been analysed in ^[20].

Properties of ${\mathscr T}_x$ with nontrivial jumps are less known, see ^[7,21,22], where a single state model with independent exponentially distributed jumps, $\log(1+Y_n)\sim \mathrm{Exp}(b),$ is being studied. Some solutions for a jump-diffusion process are presented in ^[23,24,25], the martingale methods are developed by ^[26] (see the review in ^[27]). For applications of jump-diffusion processes to models of neuronal activity see ^[28].

In this paper, we generalise the results of ^[7] on a two-state model with positive independent random jumps $Z_n = \log(1+Y_n),$ $n\geq1,$ having the alternating exponential distributions, $\mathrm{Exp}(b_0)$ and $\mathrm{Exp}(b_1),$ $b_0, b_1 > 0,$ which corresponds to the (alternating) Pareto distributions of the second kind (Lomax distributions) of $Y_n$, used in economics and actuarial science, see ^[29,30],

In Section 2, we treat the problem in a general setting. For the case of exponentially distributed jumps, explicit formulae for the moment generating function of ${\mathscr T}_x, \; x > 0,$ are obtained. The firing probabilities and the mean values of the firing time are also studied.

Section 3 concerns the limit behaviour of ${\mathscr T}_x$ under small frequent stimuli. In Section 4 we give an overview of the single-state case, including the limit behaviour under the parameters' scaling similar to that described in Section 3.

2.   Firing probabilities and the first passage time of the jump-telegraph process

Let ${\mathscr T}_x^{(i)}$ be the the first passage time of $X_i(t),$ (1.1)-(1.2), through the positive threshold $x,$

By definition, we set ${\mathscr T}_x^{(i)}|_{x < 0}\equiv0, \; i\in\{0, 1\}$.

Since $c_0, c_1\leq0$, the process ${\mathbb X}$ exceeds the threshold x$x, \; x > 0,$ just by jumping. Conditioning on the first switching we have the following identities in law:

see the definition of $X(t),$ (1.2). Here $T^{(0)}/\; Y^{(0)}$ and $T^{(1)}/\; Y^{(1)}$ are the first holding time / the first stimulus amplitude at the states ${\varepsilon}(0) = 0$ and ${\varepsilon}(0) = 1$ respectively.

Denote by $\phi_i(x) = \phi_i(x; q)$ the Laplace transform of ${\mathscr T}_x^{(i)},$

By definition, $0\leq\phi_i(x)\leq1\; \forall x,$ and $\phi_0(x)\equiv1, \; \phi_1(x)\equiv1,$ if $x < 0.$ Integrating by parts in (2.2), we have

where ${{\rm{e}}}_q$ is an exponentially distributed random variable, $\mathrm{Exp}(q),$ independent of ${\varepsilon}$ and $\{Y_n\}_{n\geq1}.$

Assuming that the sequential jumps $Y_n$ have the distributions $g_0$ and $g_1,$ alternating together with the alternating distributions $\pi_0$ and $\pi_1$ of holding times, identity (2.1) can be written as

Equations (2.3) are equivalent to

Here

denotes the conditional survivor function of the stimulus amplitude under the state ${\varepsilon} = i, \; i\in\{0, 1\}.$

In what follows, assume excitatory inputs to be positive and exponentially distributed, that is

with the corresponding density functions

We try to find the solution $\vec\phi = (\phi_0, \; \phi_1)'$ of (2.4) in the form

with the indefinite coefficients ${\mathbf A}_k = (A_{0k}, \; A_{1k})', \; {\mathbf A}_k\neq\mathit{\boldsymbol{0}},$ and $\xi_k, \; \xi_k\neq b_0, b_1$, $k = 1, \ldots, N.$ Substituting (2.7) and (2.5)-(2.5) into (2.4), we obtain the following algebraic system:

where

is the Laplace transform of the distribution of the holding time.

From (2.8), we get the following linear equations for the indefinite coefficients $\xi_k$ and $A_{ik},$ $i\in\{0, 1\}, \; k = 1, \ldots, N$:

and

The $k$-th system of (2.10) has a nontrivial ($A_{ik}\neq0$) solution, if and only if $\xi_k = \xi_k(q)$ is the root of the equation

Since the mappings $\xi\to\widehat\pi_0(q-c_0\xi)$ and $\xi\to\widehat\pi_1(q-c_1\xi), \; \xi > 0,$ (with negative $c_0$ and $c_1$) are positive decreasing functions and $\widehat\pi_0(q)\cdot\widehat\pi_1(q) < 1\; \forall q > 0,$ equation $(2.11)_q$ has exactly two real positive roots $\xi_1 = \xi_1(q), \; \xi_1 < b_0\wedge b_1,$ and $\xi_2 = \xi_2(q),$ $\xi_2 > b_0\vee b_1,$ see Fig. 1.

Hence, the moment generating function $\vec\phi(x; q)$ is defined by (2.7) with $N = 2,$

The corresponding coefficients ${\mathbf A}_1 = (A_{01}, \; A_{11})'$ and ${\mathbf A}_2 = (A_{02}, \; A_{12})'$ are determined by system (2.9)-(2.10), which splits onto two dual linear systems

and

After easy algebra, one can obtain the following explicit formulae

where the following notations

are used.

To study the firing probabilities, ${\mathbb P}\left\{ {\mathscr T}_x^{(0)} < \infty\right\}, \; {\mathbb P}\left\{ {\mathscr T}_x^{(1)} < \infty\right\},$ and the mean value of firing time, ${\mathbb E}\left[{\mathscr T}_x^{(0)}\right], \; {\mathbb E}\left[{\mathscr T}_x^{(1)}\right],$ we are interested to analyse the limits of the moment generating function and its derivative under $q\downarrow0,$

To do this, keeping in mind (2.12), we need $\xi_k(q)|_{q\downarrow0}$ and $\dfrac{ {{\rm{d}}} \xi_k(q)}{ {{\rm{d}}} q}|_{q\downarrow0},$ $k = 1, 2,$ where $\xi_1 = \xi_1(q),$ $\xi_1(q) < b_0\wedge b_1$ and $\xi_2 = \xi_2(q), \; \xi_2(q) > b_1\vee b_1, \; q > 0,$ are the two branches of (positive) roots of $(2.11)_q$.

The firing probabilities ${\mathbb P}\left\{ {\mathscr T}_x^{(i)} < \infty\right\} = \lim_{q\downarrow0}\phi_i(x; q), \; i\in\{0, 1\},$ are presented by the following proposition.

Proposition 2.1. Let the mean value of the holding time intervals $T_n, \; n\geq1,$ exists, that is

● If

then the limits $\xi_* = \lim_{q\downarrow0}\xi_1(q)$ and $\xi^* = \lim_{q\downarrow0}\xi_2(q)$ exist and are positive.

The firing probabilities are given by

where $A_{ik}^* = A_{ik}(\xi_*, \xi^*), \; i\in\{0, 1\}, k = 1, 2,$ are defined by (2.13)-(2.16) with $\xi_1 = \xi_*, \; \xi_2 = \xi^*$.

● Otherwise, if

then the firing occurs a. s.

Proof. Note that $\widehat\pi_0(q-c_0\xi), \; \widehat\pi_1(q-c_1\xi), \; c_0, c_1 < 0,$ are positive decreasing convex functions of $q > 0$ and of $\xi > 0.$

Since $\widehat\pi_0(0) = \widehat\pi_1(0) = 1,$ then $\xi = 0$ is the root of equation $(2.11)_0$. The other roots of equation $(2.11)_0$ depend on the relation between the values of the derivative on $\xi$ at point $\xi = 0$ of the sides of this equation. The derivatives of the RHS and of the LHS of $(2.11)_q$ are given by

respectively, see (2.18).

We have two distinct situations.

● Let (2.19) holds, that is $-\left(\dfrac{1}{b_0}+\dfrac{1}{b_1}\right) > c_0 {\mathbb E}_0[T]+c_1 {\mathbb E}_1[T].$

Since $\widehat\pi_0(-c_0\xi)\cdot\widehat\pi_1(-c_1\xi), \; \xi > 0,$ is the positive decreasing convex function, $\xi\to(1-\xi/b_0)\cdot(1-\xi/b_1)$ is convex, and

equation $(2.11)_0$ has explicitly two positive roots: $\xi_* < b_0\wedge b_1, \; \xi^* > b_0\vee b_1,$ see Fig. 2 (left). Functions $q\to\widehat\pi_0(q-c_0\xi)$ and $q\to\widehat\pi_1(q-c_1\xi)$ are monotone decreasing. Hence, the roots $\xi_1(q), \; \xi_2(q)$ of $(2.11)_q,$ $q > 0,$ are monotone functions. Further, for any positive $q$ we have

and

Equalities (2.20) follow by passing to limit in (2.12)-(2.17).

● On the contrary, let (2.21) holds, that is $-\left(\dfrac{1}{b_0}+\dfrac{1}{b_1}\right) \leq c_0 {\mathbb E}_0[T]+c_1 {\mathbb E}_1[T].$

In the case of the strict inequality,

equation $(2.11)_0$ has only one positive root

$\xi^*,$ $\xi^* > b_0\vee b_1,$ see Fig. 2 (right), and

In this case, see (2.17),

Hence, by (2.13)-(2.16) we have

Therefore,

and

If the equality holds, $-\left(\dfrac{1}{b_0}+\dfrac{1}{b_1}\right) = c_0 {\mathbb E}_0[T]+c_1 {\mathbb E}_1[T],$ then we have

and

Therefore, since $\widehat\pi_0(-c_0\xi)\widehat\pi_1(-c_1\xi)$ and $\left(1-\frac{\xi}{b_0}\right)\left(1-\frac{\xi}{b_1}\right)$ are decreasing convex functions, the same result occurs: (2.25)-(2.26), and then (2.22).

The mean value of the firing time can be obtained in a similar way. We need some auxiliary results.

Let $\xi_1(q), \; \xi_2(q)$ be the two branches of positive roots of $(2.11)_q, \; q > 0,$ and condition (2.21) holds. By proposition 2.1, $0 < \xi_1(q) < b_0\vee b_1 < \xi_2(q)$ and

Let coefficients $A_{ik} = A_{ik}(\xi_1, \xi_2;\; q)$ be defined by (2.13)-(2.17).

Lemma 2.2. Let (2.21) be satisfied. The following limit relations hold:

and

where

Proof. Substitute $\xi_1 = \xi_1(q)$ into $(2.11)_q$. Formula (2.27) follows from (2.23)-(2.24) by differentiating in $(2.11)_q.$ The derivative in $q$ at $q\downarrow0$ gives

Under condition (2.21), $\lim_{q\downarrow0}\xi_1(q) = 0.$ By definition (2.17), it follows that

and

With this in mind, by (2.13)-(2.14) you can get

and

Substituting (2.33)-(2.37) into

where $\xi_k'(0): = {{\rm{d}}}\xi_k(q)/ {{\rm{d}}} q|_{q\downarrow0}, \; k = 1, 2,$ one can get Eqns (2.28)-(2.29). Eqns (2.30)-(2.31) follow similarly.

Proposition 2.3. If (2.21) holds, then the mean firing time $\mathbf M(x) = \left({\mathbb E}\left[ {\mathscr T}_x^{(0)}\right], \; {\mathbb E}\left[ {\mathscr T}_x^{(1)}\right]\right)$ is finite and is given by the entries

where $\sigma, \; B_0(\xi^*), \; B_1(\xi^*)$ and $\xi^* = \lim_{q\downarrow0}\xi_2(q)$ are defined in Lemma 2.2.

A numerical example with various values of jump amplitudes is depicted in Fig. 3: the average firing time increases when up jumps vary from large to small.

Proof. Since $\mathbf M(x) = -\lim_{q\downarrow0}\dfrac{ {{\rm{d}}}\vec\phi(x; q)}{ {{\rm{d}}} q},$ by (2.12)

By (2.25) and (2.26) we have ${\mathbf A}_{1}|_{q\downarrow0} = \mathbf{1}, \; {\mathbf A}_{2}|_{q\downarrow0} = \mathbf{0},$ $\xi_1|_{q\downarrow0} = 0, \; \xi_2|_{q\downarrow0} = \xi^*$ and $\xi_1'|_{q\downarrow0} = \sigma.$ Therefore,

which by Lemma 2.2 gives (2.38)-(2.39).

Remark 2.1. Firing time distribution in the markovian case. Let the holding times be exponentially distributed with alternating mean values ${\lambda}_0^{-1}, \; {\lambda}_1^{-1}.$ In this case, the pair $\langle X(t), {\varepsilon}(t)\rangle, \; t\geq0,$ is the Markov process.

Since,

functions $f_0$ and $f_1,$ see (2.17), are defined by

and equation $(2.11)_q$ becomes

In this case formulae (2.38)-(2.39) for mean firing times hold with

where $\xi^* = \lim_{q\downarrow0}\xi_2(q).$

Formulae (2.38)-(2.39) for mean firing times can be simplified to an explicit form also in the case an alternating compound Poisson process, that is if $c_0 = c_1 = 0,$ which gives a nice additional result.

Proposition 2.4. Let the telegraph component vanish, $c_0 = c_1 = 0.$

In this case condition (2.21) always holds. The mean firing times of ${\mathscr T}_x$ are given by

and

where $\sigma,$ see (2.27), is simplified to

Proof. The moment generating function $\phi$ is given by (2.12), where $\xi_1(q), \; \xi_2(q)$ are the two (positive) roots of the equation $(2.11)_q$ with $c_0 = c_1 = 0:$

Explicitly,

Formulae for the coefficients $A_{ik}, \; i\in\{0, 1\}, k = 1, 2,$ can be simplified. First, by (2.17) we have

Further, by (2.13) and (2.14)

and

where $\Delta_0 = \dfrac{b_1-b_0+2b_0\widehat\pi_1(q)}{D}.$

Similarly, by (2.15) and (2.16)

where $\Delta_1 = \dfrac{b_0-b_1+2b_1\widehat\pi_0(q)}{D}.$

After easy algebra one can obtain the explicit formulae for the moment generating functions of ${\mathscr T}_x$:

Further, by (2.42) $\xi^* = \lim_{q\downarrow0}\xi_2(q) = 2b;$ by (2.43) $f_0(\xi^*; 0) = -b_0$. Similarly, $f_1(\xi^*; 0) = -b_1$ and by (2.32)

Under these simplifications, formulae (2.38)-(2.39), Proposition 2.3, become (2.40)-(2.41).

3.   Firing time under small frequent stimuli

Consider the case of small frequent stimuli. We assume that the mean values of jumps, $b_0^{-1}, \; b_1^{-1},$ and the holding times, $T_n,$ consistently tend to zero. We are interested to study the asymptotical behaviour of the firing time under these circumstances. Let's set the exact assertion.

Let the parameters of stochastic stimulation of a neuron be scaled as follows: first, the mean stimuli amplitudes consistently tend to zero,

second, the holding time intervals tend to zero. More precisely, assume that for any fixed positive $q$

such that the derivatives of the moment generating functions $\widehat\pi_0$ and $\widehat\pi_1$ exist and vanish:

Assume the convergence rates at (3.1) to be comparable,

convergence rates at (3.1), (3.3)-(3.4) to be consistent as follows:

and

Coefficients $v_0, \; v_1$ describe an additional positive trend arising in the jump-telegraph process due to small frequent positive jumps.

Let

To analyse the comportment of the roots $\xi_1$ and $\xi_2$ of $(2.11)_q$ we will use the following decomposition of the moment generating functions of the holding times distributions:

where by (3.4)

Condition (3.7) provides the uniform in $\xi$ convergence:

Theorem 3.1. Let $b_0, \; b_1\to\infty,$ and the holding times be asymptotically zero, such that conditions (3.1)-(3.6) and (3.7) met.

If $\kappa\in(0, \; +\infty],$ then for any $x, \; x > 0,$

where

If $\kappa\leq0,$ then

Proof. To analyse the asymptotical behaviour of ${\mathscr T}_x$ we need to evaluate the comportment of the positive roots $\xi_1(q), \xi_2(q), \; \xi_1(q) < \xi_2(q),$ of equation $(2.11)_q$.

It turns out, the behaviour of the smaller root $\xi_1, \; \xi_1 < b_0\wedge b_1,$ depends on the sign of $\kappa,$ (3.8).

Due to (3.9) equation $(2.11)_q$ takes the form

which can be rewritten as

Here $A_0$ and $A_1$ are constants (depending only on $q$),

and $A_2 = A_2(\xi)$ is given by

By (3.10) and (3.6)

By (3.6) and (3.1)-(3.2), $b_0A_1$ converges to $\kappa,$ (3.8),

Next,

Since $\xi = \xi_1 < b_0\wedge b_1,$ the terms $\xi m_0(q)$ and $\xi m_1(q)$ by (3.6) are uniformly bounded. Therefore, by (3.10),

If $\kappa > 0,$ the smaller root $\xi_1$ of (3.11) has the following limit:

By (3.12), (3.13) and (3.14) this limit is positive and finite:

Further, under this scaling

Under the scaling (3.1), the greater root $\xi_2,$ $\xi_2 > b_0\vee b_1,$ always goes to infinity, $\xi_2\to\infty;$ moreover, $f_0(\xi_2)/b_1$ and $f_1(\xi_2)/b_0$ are finite.

As a consequence, $A_{01}$ and $A_{11},$ which are defined by (2.13) and (2.15), converge to 1 and $A_{02}$ and $A_{12}$ are bounded.

Summarising, we obtain

which means

If the limit $\lim [b_0A_1] = \kappa$ in (3.13) is not positive, $\kappa\leq0,$ then (see (3.15))

which corresponds to

and

Remark 3.1. The result of Theorem 3.1 can be interpreted as the behaviour of two types of neurons : if $\kappa > 0,$ then the scaled model corresponds to the so called a tonically discharging cell, a phasic cell appears when $\kappa\leq0,$ that is ${\mathscr T}_x\to\infty,$ the firing rate drops to zero, see ^[31].

Remark 3.2. In the markovian case, that is if $\widehat\pi_0(q) = {\lambda}_0/(q+ {\lambda}_0), \; \widehat\pi_1(q) = {\lambda}_1/(q+ {\lambda}_1),$ conditions (3.2)-(3.7) hold when

The crucial parameter $\kappa$ becomes

4.   Single-state homogeneous model

The model of neural activity based on a jump-telegraph process, see (1.1), (1.4), which is studied in Sections 2 and 3 can be simplified, restricting to the case with one state. Consider the particular case of the neural model (1.4) based on the single-state symmetric process

with independent positive exponentially distributed stimuli amplitudes $Y_n, \; Y_n\sim \mathrm{Exp}(b),$ $b > 0;$ the independent inter-arrival times $\{T_n\}_{n\geq1},$ are identically distributed with the density function $\pi(t)$. When $T_n$ are exponentially distributed, such a model has been studied in detail by ^[7] and ^[21].

The moment generating function $\phi(x) = {\mathbb E}\left[\exp(-q {\mathscr T}_x)\right]$ of the first passage time ${\mathscr T}_x$ is given by (2.12),

where $A_1 = A_{01} = A_{11}, \; A_2 = A_{02} = A_{12}$ are defined by (2.13)-(2.17), $\xi_1 = \xi_1(q)$ and $\xi_2 = \xi_2(q)$ are the two branches of positive roots of $(2.11)_q.$

For the single-state process $X(t),$ defined by (4.1), equation $(2.11)_q$ is simplified to

More precisely, $\xi_1, \; \xi_1 < b,$ is the positive root of $\widehat \pi(q-c\xi) = 1-{\xi/b},$ and $\xi_2, \; \xi_2 > b,$ is the root of $\widehat \pi(q-c\xi) = {\xi/b}-1.$ In this case, functions $f_0$ and $f_1,$ which are defined by (2.17), coincide,

that is,

By (4.2) and (2.13)-(2.16)

and the moment generating function $\phi$ (depending only on $\xi_1)$ is given by

where $\xi = \xi_1(q), \; 0 < \xi_1(q) < b, \; q > 0,$ is the root of

Function $\widehat\pi = \widehat\pi(q), \; q\geq0,$ is positive, convex and decreasing, hence the root of (4.4) exists and function $\xi_1(q), \; q\geq0,$ increases,

Therefore, $\lim_{q\downarrow0}\xi_1(q)$ exists and, by Proposition 2.1,

Further, the firing probability ${\mathbb P}\{ {\mathscr T}_x < \infty\} = \lim_{q\downarrow0}\phi(x; q)$ is given by

In the case $c {\mathbb E}[T]+b^{-1} > 0,$ by differentiating in (4.4) (see also (2.27)) one can obtain

The mean firing time is finite: by (4.3) and (4.5) one can obtain

This result is in concordance with Proposition 2.3. In this case, by (2.27),

and by (2.32),

which by (2.38)-(2.39) gives (4.6).

In particular, consider the model (4.1) defined by the compound Poisson process with the (negative) drift $c,$ that is, let the inter-switching times of model (4.1) be exponentially distributed, $\pi(t) = {\lambda}\exp(-{\lambda} t).$ Now, formula (4.3) holds with $\xi, \; 0 < \xi < b,$ which is the unique positive root of the equation (4.4):

We have

where $\widetilde q: = q+ {\lambda}-bc$. Due to (4.3), the moment generating function $\phi$ is given by

which coincide with [7,(39)]. By applying the inverse Laplace transform $\mathscr L^{-1}_{q\to t}$ to (4.8) one can obtain the firing density $f_{ {\mathscr T}_x}(t)$. Formula [32,2.2.5-18] applied to $\exp(-\xi(q)x)$ with $\xi(q)$ defined by (4.7) shows

where $a = x/(-2c), \; z = 2\sqrt{-bc {\lambda}}.$ After easy algebra, from (4.8) we obtain

with $w: = 2\sqrt{ {\lambda} bt(x-ct)}.$ Formula (4.9) was derived in [7, Theorem 3.1] using another technique.

Firing probability, ${\mathbb P}\{ {\mathscr T}_x < \infty\},$ and moments of ${\mathscr T}_x$ can be also obtained: if ${\lambda}+bc > 0,$ then (2.21) holds and

If ${\lambda}+b c\leq0,$ then ${\mathbb E}\left[{\mathscr T}_x\right] = +\infty.$ This coincides with the known result, see [7, Proposition 4.2].

The limit behaviour under small frequent stimuli, Section 3, in the case of the single-state model, also looks simple.

In this case, condition (3.1) corresponds to $b\to\infty$ and $\beta = 1$, see (3.5); (3.6)-(3.7) follows, if ${\lambda}\to+\infty$ and $\dfrac{ {\lambda}}{b}\to v.$

By (4.3)

where $\xi_1 = \xi_1(q)$ is the positive root, $0 < \xi_1(q) < b,$ of

One can see that for any $q > 0$ under this scaling

and for any $x > 0$

Therefore,

5.   Concluding remarks

The main goal of this paper is to study the stochastic model based on two states/phases of the nerve cell, alternating at random times of exponential excitatory inputs. The corresponding single-phase Stein's model is well known and presented in detail, see ^[3,5,10]. Our model generalises and modifies the two-phase cell cycle model, presented by ^[10,11]. In this paper, we have obtained the explicit formulae for firing probability and the mean firing time, under the certain necessary condition. The asymptotical behaviour of the firing time under small frequent stimuli has been also presented. The known results of the single-state homogeneous model, ^[7], follow as a special case.

Since the real activity of neurons depends on the current state/phase of the organism, the proposed model, based on two alternating patterns, fits well with a naive understanding of this issue. The structure of this model can serve as a guide for practitioners: it would be interesting to discover this two-phase phenomenon of the behaviour of neurons in an experiment.

Acknowledgments

The author thanks two anonymous referees for their helpful comments that improved the paper.

Conflict of interest

The author declares no conflicts of interest in this paper.