^{1}, Jan A. Freund

^{1,5}, Epaminondas Rosa Jr.

^{1}, Paul H. Bryant

^{3}, Hans A. Braun

^{4}and Ulrike Feudel

^{5}

### Abstract

The response of a four-dimensional mammalian cold receptor model to different implementations of noise is studied across a wide temperature range. It is observed that for noisy activation kinetics, the parameter range decomposes into two regions in which the system reacts qualitatively completely different to small perturbations through noise, and these regions are separated by a homoclinic bifurcation. Noise implemented as an additional current yields a substantially different system response at low temperature values, while the response at high temperatures is comparable to activation-kinetic noise. We elucidate how this phenomenon can be understood in terms of state space dynamics and gives quantitative results on the statistics of interspike interval distributions across the relevant parameter range.

Mathematical models which mimic the electrical properties of nerve membranes on the basis of physiological mechanisms are an important tool in modern neuroscience. The advent of this modeling approach dates back to the groundbreaking work of Hodgkin and Huxley on the squid giant axon, which was published in 1952. While simplistic models had long been able to reproduce the shape of electrical discharges of membranes more or less accurately, mechanism-based models hold the prospect of giving away experimentally accessible insights into the workings of the membrane. Since natural biological systems are inherently noisy, it is an important task to include stochastic components into the models. A significant source of stochastic fluctuations is related to intrinsic properties of the membrane such as opening and closing mechanisms of ion channels, thereby giving rise to intrinsic dynamical noise. Additionally, one faces here a disadvantage of real electrophysiological recordings—the data are also subject to noise sources which are not rooted in the dynamics of or around the cell, but which are induced by external perturbations and are known as measurement noise. Finding the correct or, in a given situation, most appropriate way of introducing noise into a model is still a hard problem. For instance, it has been demonstrated that different choices of which of the model’s equations will be subjected to noise can lead to different qualitative behaviour. It is the aim of this work to elucidate the mechanisms behind such diverse responses to different, physiologically justified, noise implementations. For our study, we use a four-dimensional model of the nerve membrane, which had originally been developed to mimic signal transduction in mammalian cold receptors. A particular advantage of this choice is that it is a very versatile neuronmodel in the sense that depending on the temperature, it exhibits a large variety of different firing patterns. This feature has made the model an often-used tool for the simulation of different kinds of neurons of the central and peripheral nervous systems. We study noise acting on the level of ion channels and compare the results to noise acting as a stochastic synaptic input, elucidating the dynamical backbone behind the different responses to different noise sources and give quantitative results on the statistical distribution of intervals between successive spikes.

The authors thank Quinton Skilling for help with simulations and graphs. Christian Finke was supported by Grant No. FE 359/10 of the German Science Foundation (DFG).

I. INTRODUCTION

II. THE MODEL

III. STATE SPACE DYNAMICS

IV. INTERSPIKE INTERVAL HISTOGRAMS

V. DISCUSSION

## Figures

Attractor bifurcation diagrams of interspike intervals vs. temperature. The ISIs correspond to the Poincaré return time for a surface of section at *V* = −20 mV. (a) The deterministic scenario; (b) Gaussian white noise of intensity *D* = 0.5 is implemented only in the voltage variable *V*; (c) Gaussian white noise of intensity *D* = 0.00002 is implemented only in the slow repolarizing variable ; (d) Gaussian white noise of intensity *D* = 0.00002 added to both activation variables *a _{r} * and

*a*. We use different noise realizations in every variable. To help the eye, the parametric locus of the deterministic homoclinic bifurcation is indicated by a vertical arrow in each panel.

_{sd}Attractor bifurcation diagrams of interspike intervals vs. temperature. The ISIs correspond to the Poincaré return time for a surface of section at *V* = −20 mV. (a) The deterministic scenario; (b) Gaussian white noise of intensity *D* = 0.5 is implemented only in the voltage variable *V*; (c) Gaussian white noise of intensity *D* = 0.00002 is implemented only in the slow repolarizing variable ; (d) Gaussian white noise of intensity *D* = 0.00002 added to both activation variables *a _{r} * and

*a*. We use different noise realizations in every variable. To help the eye, the parametric locus of the deterministic homoclinic bifurcation is indicated by a vertical arrow in each panel.

_{sd}Deterministic time series of the membrane voltage at different temperature values. (a) Tonic firing at *T* = 4 °C; (b) Chaotic bursting below the homoclinic bifurcation at *T* = 10 °C; (c) Chaotic bursting beyond the homoclinic bifurcation at *T* = 11.5 °C; and (d) Period-2 bursting at *T* = 25 °C. The homoclinic bifurcation is located at *T* = 10.7 °C. In panels (b)–(d), subthreshold oscillations are clearly visible.

Deterministic time series of the membrane voltage at different temperature values. (a) Tonic firing at *T* = 4 °C; (b) Chaotic bursting below the homoclinic bifurcation at *T* = 10 °C; (c) Chaotic bursting beyond the homoclinic bifurcation at *T* = 11.5 °C; and (d) Period-2 bursting at *T* = 25 °C. The homoclinic bifurcation is located at *T* = 10.7 °C. In panels (b)–(d), subthreshold oscillations are clearly visible.

State space dynamics of the Huber-Braun model at (a) *T* = 4 °C (deterministic period-one solution) and (b) *T* = 25 °C (deterministic period-two solution). The deterministic solutions are shown as a solid black line, the grey trajectories show the situation when the voltage variable *V* is subjected to additive Gaussian white noise of intensity *D* = 0.5. The circle indicates the position of the saddle equilibrium. The corresponding voltage traces are depicted in Figs. 6(a) and 6(b).

State space dynamics of the Huber-Braun model at (a) *T* = 4 °C (deterministic period-one solution) and (b) *T* = 25 °C (deterministic period-two solution). The deterministic solutions are shown as a solid black line, the grey trajectories show the situation when the voltage variable *V* is subjected to additive Gaussian white noise of intensity *D* = 0.5. The circle indicates the position of the saddle equilibrium. The corresponding voltage traces are depicted in Figs. 6(a) and 6(b).

State space dynamics of the Huber-Braun model at (a) *T* = 4 °C (deterministic period-one solution) and (b) *T* = 25 °C (deterministic period-two solution). The deterministic solutions are shown as a solid black line, the grey trajectories show the situation when the slow repolarizing variable is subjected to additive Gaussian white noise of intensity *D* = 0.00002. The circle indicates the position of the saddle equilibrium. The corresponding voltage traces are depicted in Figs. 6(c) and 6(d).

State space dynamics of the Huber-Braun model at (a) *T* = 4 °C (deterministic period-one solution) and (b) *T* = 25 °C (deterministic period-two solution). The deterministic solutions are shown as a solid black line, the grey trajectories show the situation when the slow repolarizing variable is subjected to additive Gaussian white noise of intensity *D* = 0.00002. The circle indicates the position of the saddle equilibrium. The corresponding voltage traces are depicted in Figs. 6(c) and 6(d).

State space dynamics of the Huber-Braun model at *T* = 10 °C. The position of the saddle, indicated by a circle, is shown in relation to the noisy attractor of the system when it is subjected to Gaussian white noise of intensity *D* = 0.00002 in . In this parameter regime close to the homoclinic bifurcation, the effects of current or conductance noise do not differ substantially from the deterministic scenario and practically lead to the same attractor structure.

State space dynamics of the Huber-Braun model at *T* = 10 °C. The position of the saddle, indicated by a circle, is shown in relation to the noisy attractor of the system when it is subjected to Gaussian white noise of intensity *D* = 0.00002 in . In this parameter regime close to the homoclinic bifurcation, the effects of current or conductance noise do not differ substantially from the deterministic scenario and practically lead to the same attractor structure.

Noisy time series of the membrane voltage at different temperature values, corresponding to the state space diagrams in Figs. 3 and 4. (a) Current noise of intensity *D* = 0.5 at *T* = 4 °C and (b) at *T* = 25 °C (cf. Fig. 3); (c) Conductance noise of intensity *D* = 0.00002 at *T* = 4 °C and (d) at *T* = 25 °C (cf. Fig. 4).

Noisy time series of the membrane voltage at different temperature values, corresponding to the state space diagrams in Figs. 3 and 4. (a) Current noise of intensity *D* = 0.5 at *T* = 4 °C and (b) at *T* = 25 °C (cf. Fig. 3); (c) Conductance noise of intensity *D* = 0.00002 at *T* = 4 °C and (d) at *T* = 25 °C (cf. Fig. 4).

Histogram plots showing the number of interspike intervals on a logarithmic scale versus the interspike intervals (a) at *T* = 10 °C and *T* = 11.5 °C, (b) at *T* = 8 °C and *T* = 25 °C, and (c) at *T* = 4 °C and *T* = 30 °C for conductance noise in of intensity *D* = 0.00002. The temperature values have been chosen in a way that the deterministic system is in a chaotic mode before and after the homoclinic bifurcation in panel (a) and in a tonic firing mode before and after the homoclinic bifurcation in panel (c). The statistics have been taken for 20 000 interspike intervals and a bin width of 0.02.

Histogram plots showing the number of interspike intervals on a logarithmic scale versus the interspike intervals (a) at *T* = 10 °C and *T* = 11.5 °C, (b) at *T* = 8 °C and *T* = 25 °C, and (c) at *T* = 4 °C and *T* = 30 °C for conductance noise in of intensity *D* = 0.00002. The temperature values have been chosen in a way that the deterministic system is in a chaotic mode before and after the homoclinic bifurcation in panel (a) and in a tonic firing mode before and after the homoclinic bifurcation in panel (c). The statistics have been taken for 20 000 interspike intervals and a bin width of 0.02.

The distribution of interspike intervals at *T* = 10 °C and *T* = 11.5 °C in the deterministic system.

The distribution of interspike intervals at *T* = 10 °C and *T* = 11.5 °C in the deterministic system.

The data points of the exponentially decaying flank of interspike intervals at *T* = 4 °C (cf. Fig. 7(c)) together with the exponential fit (solid line).

The data points of the exponentially decaying flank of interspike intervals at *T* = 4 °C (cf. Fig. 7(c)) together with the exponential fit (solid line).

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