# The stochastic Fitzhugh–Nagumo neuron model in the excitable regime embeds a leaky integrate-and-fire model

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## Abstract

In this paper, we provide a complete mathematical construction for a stochastic leaky-integrate-and-fire model (LIF) mimicking the interspike interval (ISI) statistics of a stochastic FitzHugh–Nagumo neuron model (FHN) in the excitable regime, where the unique fixed point is stable. Under specific types of noises, we prove that there exists a global random attractor for the stochastic FHN system. The linearization method is then applied to estimate the firing time and to derive the associated radial equation representing a LIF equation. This result confirms the previous prediction in Ditlevsen and Greenwood (J Math Biol 67(2):239–259, 2013) for the Morris-Lecar neuron model in the bistability regime consisting of a stable fixed point and a stable limit cycle.

## Keywords

FitzHugh–Nagumo model Excitable regime Leaky integrate-and-fire model Random attractor Stationary distribution## Mathematics Subject Classification

60GXX 92BXX## 1 Introduction

Mathematical modeling has emerged as an important tool to handle the overwhelming structural complexity of neuronal processes and to gain a better understanding of their functioning from the dynamics of their model equations. However, the mathematical analysis of biophysically realistic neuron models such as the 4-dimensional Hodgkin–Huxley (HH) (1952) and the 2-dimensional Morris–Lecar (ML) (1981) equations is difficult, as a result of a large parameter space, strong nonlinearities, and a high dimensional phase space of the model equations. The search for simpler, mathematically tractable (small parameter space, weaker nonlinearities, low dimensional phase space) neuron models that still capture all, or at least some important dynamical behaviors of biophysical neurons (HH and ML) has been an active area of research.

The efforts in this area of research have resulted in easily computable neuron models which mimic some of the dynamics of biophysical neuron models. One of the resulting models is the 2-dimensional FitzHugh–Nagumo (FHN) neuron model (FitzHugh 1961). The FHN model has been so successful, because it is at the same time mathematically simple and produces a rich dynamical behavior that makes it a model system in many regards, as it reproduces the main dynamical features of the HH model. In fact, the HH model has two types of variables, and each type then is combined into a single variable in FHN: The (*V*, *m*) variables of HH correspond to the *v* variable in FHN, whose fast dynamics represents excitability; the (*h*, *n*) variables correspond to the *w* variable, whose slow dynamics represents accommodation and refractoriness.

The fact that the FHN model is low dimensional makes it possible to visualize the solution and to explain in geometric terms important phenomena related to the excitability and action potential generation mechanisms observed in biological neurons. Of course, this comes at the expense of numerical agreement with the biophysical neuron models (Yamakou 2018). The purpose of the model is not a close match with biophysically realistic high dimensional models, but rather a mathematical explanation of the essential dynamical mechanism behind the firing of a neuron. Moreover, the analysis of such simpler neuron models may lead to the discovery of new phenomena, for which we may then search in the biological neuron models and also in experimental preparations.

There is, however, an even simpler model than FHN, the leaky integrate-and-fire model (LIF). This is the simplest reasonable neuron model. It only requires a few basic facts about nerve cells: they have membranes, they are semipermeable, and they are polarizable. This suffices to deduce a circuit equivalent to that of the membrane potential of the neuron: a resistor-capacitor circuit. Such circuits charge up slowly when presented with a current, cross a threshold voltage (a spike), then slowly discharge. This behavior is modeled by a simple 1D equation together with a reset mechanism: the leaky integrate-and-fire neuron model equation (Gerstner and Kistler 2002). Combining sub-threshold dynamics with firing rules has led to a variety of 1D leaky integrate-and-fire descriptions of a neuron with a fixed membrane potential firing threshold (Gerstner and Kistler 2002; Lansky and Ditlevsen 2008) or with a firing rate depending more sensitively on the membrane potential (Pfister et al. 2006). In contrast to \(n-\)dimensional neuron models, \(n\ge 2\), such as the HH, ML, and FHN models, the LIF class of neuron models is less expensive in numerical simulations, which is an essential advantage when a large network of coupled neurons is considered.

Noise is ubiquitous in neural systems and it may arise from many different sources. One source may come from synaptic noise, that is, the quasi-random release of neurotransmitters by synapses or random synaptic input from other neurons. As a consequence of synaptic coupling, real neurons operate in the presence of synaptic noise. Therefore, most works in computational neuroscience address modifications in neural activity arising from synaptic noise. Its significance can however be judged only if its consequences can be separated from the internal noise, generated by the operations of ionic channels (Calvin and Stevens 1967). The latter is channel noise, that is, the random switching of ion channels. In many papers channel noise is assumed to be minimal, because typically a large number of ion channels is involved and fluctuations should average out, and therefore, the effects of synaptic noise should dominate. Consequently, channel noise is frequently ignored in the mathematical modeling. However, the presence of channel noise can also greatly modify the behavior of neurons (White et al. 2000). Therefore, in this paper, we study the effect of channel noise. Specifically, we add a noise term to the right-hand side of the gating equations (the equation for the ionic current variable).

In the stochastic model, the deterministic fixed point is no longer a solution of the system. The fixed point necessarily needs to vary and adapt to the noise. To account for this, in the theory of random dynamical systems, the notion of a random dynamical attractor was developed as a substitute for deterministic attractors in the presence of noise. In the first part of this paper, we therefore prove that our system admits a global random attractor, for both additive and multiplicative channel noises. This can be seen as a theoretical grounding of our setting.

In Ditlevsen and Greenwood (2013), it was shown that a stochastic LIF model constructed with a radial Ornstein–Uhlenbeck process is embedded in the ML model (in a bistable regime consisting of a fixed point and limit cycle) as an integral part of it, closely approximating the sub-threshold fluctuations of the ML dynamics. This result suggests that the firing pattern of a stochastic ML can be recreated using the embedded LIF together with a ML stochastic firing mechanism. The LIF model embedded in the ML model captures sub-threshold dynamics of a combination of the membrane potential and ion channels. Therefore, results that can be readily obtained for LIF models can also yield insight about ML models. In the second part of this paper, we here address the problem to obtain a stochastic LIF model mimicking the interspike interval (ISI) statistics of the stochastic FHN model in the excitable regime, where the unique fixed point is stable. Theoretically, we obtain such a LIF model by reducing the 2D FHN model to the one dimensional system that models the distance of the solution to the random attractor as shown in the first part of the paper. In fact, we show that this distance can be approximated to the fixed point, up to a rescaling, as the Euclidean norm \(R_t\) of the solution of the linearization of the stochastic FHN equation along the deterministic equilibrium point, and hence the LIF model is approximated by the equation for \(R_t\). An action potential (a spike) is produced when \(R_t\) exceeds a certain firing threshold \(R_t\ge r_0>0\). After firing the process is reset and time is back to zero. The ISI \(\tau _0\) is identified with the first-passage time of the threshold, \(\tau _0=\inf \{t>0: R_t\ge r_0>0\}\), which then acts as an upper bound of the spiking time \(\tau \) of the original system. By defining the firing as a series of first-passage times, the 1D radial process \(R_t\) together with a simple firing mechanism based on the detailed FHN model (in the excitable regime), the firing statistics is shown to reproduce the 2D FHN ISI distribution. We also show that \(\tau \) and \(\tau _0\) share the same distribution.

The rest of the paper is organized as follows: Sect. 2 introduces the deterministic version of the FHN neuron model, where we determine the parameter values for which the model is in the excitable regime. In Sect. 3, we prove the existence of a global random attractor of the random dynamical system generated by the stochastic FHN equation; and furthermore derive a rough estimate for the firing time using the linearization method. The corresponding stochastic LIF equation is then derived in Sect. 4 and its distribution of interspike-intervals is found to numerically match the stochastic FHN model.

## 2 The deterministic model and the excitable regime

*t*, the deterministic FHN neuron model is

*I*is a constant bias current which can be considered as the effective external input current. \(0<\varepsilon :=t/\tau \ll 1\) is a small singular perturbation parameter which determines the time scale separation between the fast

*t*and the slow time scale \(\tau \). Thus, the dynamics of \(v_t\) is much faster than that of \(w_t\). \(\alpha \) and \(\beta \) are parameters.

*layer problem*associated to Eq. (2.1) (i.e., the equation obtained from Eq. (2.1) in the singular limit \(\epsilon = 0\), see Kuehn (2015) for a comprehensive introduction to slow-fast analysis), is obtained by solving \(f(v,w)=0\) for

*w*. Thus, it is given by

*v*-nullcline (the red curve in Fig. (1)). The stability of points on \(\mathcal {C}_0\) as steady states of the

*layer problem*associated to Eq. (2.1) is determined by the Jacobian scalar \((D_vf)(v,w)=1-v^2\). This shows that on the critical manifold, points with \(|v|>1\) are stable while points with \(|v|<1\) are unstable. It follows that the branch \(v_{-}^*(w)\in (-\infty ,-1)\) is stable, \(v_0^*(w)\in (-1,1)\) is unstable, and \(v_+^*(w)\in (1,+\infty )\) is stable.

*w*-nullcline (\(w=\frac{v+\alpha }{\beta }\)) at one, two or three different fixed points. We assume in this paper that \(\bigtriangleup >0\), in which case we have a unique fixed point given by

*M*) of Eq. (2.1) at the fixed point \((v_e,w_e)\) has a pair of complex conjugate eigenvalues

## 3 The stochastic model

*f*and

*g*are given in Eq. (2.1). There are two important cases: either \(h(w) = \sigma _0\) (additive channel noise) or \(h(w)=\sigma _0 w\) (multiplicative channel noise). \(\circ dB_t\) stands for the Stratonovich stochastic integral with respect to the Brownian motion \(B_t\).

*F*is dissipative in the weak sense, i.e.

*H*is globally Lipschitz continuous.

### 3.1 The existence of a random attractor

In the sequel, we are going to prove that there exists a unique solution \(\mathbf X(\cdot ,\omega ,\mathbf X_0)\) of (3.1) and the solution then generates a so-called *random dynamical system* (see e.g. Arnold 1998, Chapters 1–2).

More precisely, let \((\Omega ,\mathcal {F},\mathbb {P})\) be a probability space on which our Brownian motion \(B_t\) is defined. In our setting, \(\Omega \) can be chosen as \(C^0(\mathbb {R},\mathbb {R})\), the space of continuous real functions on \(\mathbb {R}\) which are zero at zero, equipped with the compact open topology given by the uniform convergence on compact intervals in \(\mathbb {R}\), \(\mathcal {F}\) as \(\mathcal {B}(C^0)\), the associated Borel-\(\sigma \)-algebra and \(\mathbb {P}\) as the Wiener measure. The Brownian motion \(B_t\) can then be constructed as the canonical version \(B_t(\omega ) := \omega (t)\).

*metric dynamical system*.

*A*is invariant under \(\varphi \), i.e. \(\varphi (t,\omega )A(\omega ) = A(\theta _t \omega )\), and attracts all other compact random sets \(D(\omega )\) in the pullback sense, i.e.

*d*(

*B*|

*A*) is the Hausdorff semi-distance. Such a structure is called a

*random attractor*(see e.g. Crauel et al. 1997 or Arnold 1998, Chapter 9).

The following theorem ensures that the stochastic system (3.1) has a global random pullback attractor. The proof is provided in the “Appendix”.

### Theorem 3.1

There exists a unique solution of (3.2) which generates a random dynamical system. Moreover, the system possesses a global random pullback attractor.

Theorem 3.1 shows that every trajectory would in the long run converge to the global random attractor. The structure and the inside dynamics of the global random attractor are still open issues which might help understand the firing mechanism.

### 3.2 The normal form at the equilibrium point

*F*at \(\mathbf X_e\), \(\bar{F}\) is the nonlinear term such that

### Theorem 3.2

*C*independent of

*r*such that for any \(t\ge 0\), the following estimates hold

- For additive noise$$\begin{aligned} \sup _{t\le \tau }\Vert \mathbf X_t - \mathbf X_e -\bar{\mathbf X}_t\Vert \le C \gamma (r) r. \end{aligned}$$(3.10)
- For multiplicative noise$$\begin{aligned} E\Vert \mathbf X_{t\wedge \tau } - \mathbf X_e -\bar{\mathbf X}_{t\wedge \tau }\Vert ^2 \le C \gamma ^2(r) r^2. \end{aligned}$$(3.11)

## 4 The embedded LIF model

*radial Ornstein–Uhlenbeck equation*. More precisely, we rewrite the linearized system (3.9) in the form

*M*has a pair of complex conjugate eigenvalues \(-\mu \pm i \nu \) with \(\mu = 0.0312496, \nu = 0.281378\). By transformation \(\bar{\mathbf Y}_t = Q^{-1} \bar{\mathbf X}_t\) with \(Q = \begin{pmatrix}-\nu &{}\, m_{11}+\mu \\ 0 &{}\, m_{21}\end{pmatrix}\) we obtain

### 4.1 Firing mechanism

A spike in Eq. (3.1) occurs when there is a transition of a random trajectory from the vicinity of the stable fixed point \(\mathbf X_e=(v_e,w_e)\) located on the left stable part of \(\mathcal C_0\) to its right stable part and back to the vicinity of \(\mathbf X_e\). This spike happens almost surely when a random trajectory with the starting point \(\mathbf X_0\) in the vicinity of \(\mathbf X_e\) crosses the threshold line \(v=0\). From the phase space of Eq. (3.1) (see Fig. 2), the probability of a spike increases as the starting point \(\mathbf X_0\) moves farther away from \(\mathbf X_e\).

In order to construct the firing mechanism of Eq. (4.3) matching that of Eq. (3.1), we will calculate the conditional probability that Eq. (3.1) fires given that the trajectory crosses the line \(L=\{(v_e,w): w \le w_e\}\). Denote by \(L_i = (v_e,w_e-l_i)\) with \(l_i = i\delta = i \frac{|w_e+0.453|}{20}\) for \(i=0,1,\ldots ,34\), then the distance between the equilibrium and \(L_i\) is \(l_i\). The value \(|w_e+0.453|\) can be considered as the distance between the fixed point \((v_e,w_e)\) and the separatrix (see also Fig. 1) along *L*. For a given pair (\(\sigma _0, l_i\)), a short trajectory starting in \(L_i\) was simulated from (3.1), it was recorded whether a spike occurred (crossing the threshold \(v=0\)) in the first cycle of the stochastic path around \((v_e,w_e)\). This was repeated 1000 times and we counted the ratio of the number of spikes, denoted by \(\hat{p}(l_i,\sigma _0)\), which is an estimate for the conditional probability of firing \(p(l,\sigma _0)\). The estimation was, furthermore, repeated for \(\sigma _0 = 0.001, 0.002, 0.003, 0.004, 0.005, 0.006, 0.007, 0.008, 0.009, 0.01, 0.015\).

*a*and

*b*then are estimated by using a non-linear regression from the above simulation data and are plotted in Fig. 6 for some different values of the noise amplitude \(\sigma _0 = 0.003, 0.005, 0.007, 0.009, 0.01\), and 0.015. We see that the family of estimates, \(\hat{p}\), fits the fitted curve quite well for each value of \(\sigma _0\). Regression estimates are reported in Table 1. Note that \(p(a)=1/2\), i.e.,

*a*is the distance along

*L*from \(w_e\) at which the conditional probability of firing equals one half. For all values of \(\sigma _0\), the estimate of

*a*is close to the distance along

*L*between \(w_e\) and the separatrix, which equals 0.05. In other words, the probability of firing, if the path starts at the intersection of

*L*with the separatrix, is about 1 / 2. The estimate of

*b*increases with respect to \(\sigma _0\), and the conditional probability approaches a step function as the amplitude of the noise goes to zero. A step function would correspond to the firing being represented by a first passage time of a fixed threshold.

Estimates of regression parameters for the conditional probability of firing in the original space (*a*, *b*) and in the transformed coordinates \((a^*, b^*)\) based on the additive noise \(\sigma _0\)

\(\sigma _0\) | 0.001 | 0.002 | 0.003 | 0.004 | 0.005 | 0.006 | 0.007 | 0.008 | 0.009 | 0.01 | 0.015 |
---|---|---|---|---|---|---|---|---|---|---|---|

| 0.050161 | 0.050268 | 0.049946 | 0.049760 | 0.049816 | 0.050001 | 0.049862 | 0.049411 | 0.049078 | 0.048559 | 0.046142 |

| 0.001028 | 0.002099 | 0.003192 | 0.004310 | 0.005281 | 0.006459 | 0.007478 | 0.008844 | 0.009877 | 0.011068 | 0.017722 |

\(a^*\) | 0.630282 | 0.631624 | 0.627576 | 0.625240 | 0.625935 | 0.628262 | 0.626516 | 0.620859 | 0.616673 | 0.610148 | 0.579777 |

\(b^*\) | 0.012918 | 0.026372 | 0.040106 | 0.054158 | 0.066352 | 0.081158 | 0.093960 | 0.111127 | 0.124107 | 0.139075 | 0.222676 |

*l*between (0,

*l*) and (0, 0) in \(\bar{\mathbf X}_t\) transforms to the distance

### 4.2 ISI distributions

*L*is

*r*at time

*t*, of a spike occurring in the next small time interval, given that it has not yet occurred.

*t*, we can numerically determine the density (4.8) up to any desired precision by choosing

*n*and

*M*large enough through the expression

*M*realizations of \(R_{it/n}, i = 0,1,\ldots , n\), and the integral has been approximated by the trapezoidal rule. The results are illustrated in Fig. 7 for \(\sigma _0=0.01\), using \(M = 1000, n=10\). The estimated ISI distributions from our approximate LIF models (4.3) and (4.4) with the firing mechanism compare well with the estimated ISI histogram of FHN (3.1) reset to 0 after firings.

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society. We thank the anonymous reviewers for their careful reading and useful remarks which helped to improve the quality of the manuscript.

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