On two diffusion neuronal models with multiplicative noise: The mean first-passage time properties

Chaos: An Interdisciplinary Journal of Nonlinear Science

Stiber

2018

The rate coding hypothesis is the oldest and still one of the most accepted and investigated scenarios in neuronal activity analyses. However, the actual neuronal firing rate, while informally understood, can be mathematically defined in several different ways. These definitions yield distinct results; even their average values may differ dramatically for the simplest neuronal models. Such an inconsistency, together with the importance of "firing rate", motivates us to revisit the classical concept of the instantaneous firing rate. We confirm that different notions of firing rate can in fact be compatible, at least in terms of their averages, by carefully discerning the time instant at which the neuronal activity is observed. Two general cases are distinguished: either the inspection time is synchronised with a reference time or with the neuronal spiking. The statistical properties of the instantaneous firing rate, including parameter estimation, are analyzed and compatibility with the intuitively understood concept is demonstrated.The electric discharge activity of neurons is composed of stereotyped events called action potentials or spikes. The exact timing of spikes under identical external conditions may vary from trial to trial. Since the early days of neuroscience it has been often assumed that neurons express information about their input by employing mainly the average firing rate (frequency) of spikes. However, reliable firing rate statistics can be difficult to obtain in certain experiments or even in mathematical models. The reciprocal value of the interval between consecutive spikes -known as the instantaneous firing rate -offers the traditionally employed alternative. Although the physical dimension of the instantaneous rate is compatible with the firing frequency, the averages of the two quantities differ. In this paper we reconcile this tension by pointing to the crucial role of the reference time at which we inspect the spike pattern. We describe two possible scenarios: the classical one, in which the inspection is aligned with spikes (yielding the mentioned incompatible averages), and the asynchronous one, in which the inspection time is fixed to an external reference time (and the mean instantaneous firing rate generally equals the mean firing frequency).

Section: Definitions Of Neuronal Firing Ratementioning

confidence: 99%

Statistics of inverse interspike intervals: The instantaneous firing rate revisited

Kostal

Chaos: An Interdisciplinary Journal of Nonlinear Science

Stiber

2018

“…These counterintuitive results are consequences of the presence of the reversal potentials in the model equation and we speculate that the same phenomena can be observed in other models with multiplicative noise, like the Feller or the IGBM [48]. The extension of these findings to the entire class of models with multiplicative noise will be the subject of a future work.…”

Section: Discussionmentioning

confidence: 54%

Inhibition enhances the coherence in the Jacobi neuronal model

D’Onofrio

Chaos, Solitons & Fractals

Tamborrino

2019

Self Cite

The output signal is examined for the Jacobi neuronal model which is characterized by input-dependent multiplicative noise. The dependence of the noise on the rate of inhibition turns out to be of primary importance to observe maxima both in the output firing rate and in the diffusion coefficient of the spike count and, simultaneously, a minimum in the coefficient of variation (Fano factor). Moreover, we observe that an increment of the rate of inhibition can increase the degree of coherence computed from the power spectrum. This means that inhibition can enhance the coherence and thus the information transmission between the input and the output in this neuronal model. Finally, we stress that the firing rate, the coefficient of variation and the diffusion coefficient of the spike count cannot be used as the only indicator of coherence resonance without considering the power spectrum.

“…Using Eqs. (11) and (12), in Fig. 3, we plot the CV of the ISIs generated by model (14) as a function of the excitatory rate λ for different fixed values of the inhibitory rate ω.…”

Section: B Firing Variabilitymentioning

confidence: 99%

“…Thus, several other different variants have been investigated. 11,12 The emphasis on the lower boundary, the inhibitory reversal potential, appears probably because the role of the upper boundary, the excitatory reversal potential, seems to be blurred by imposing below it a firing threshold. However, despite the existence of the threshold, the upper boundary modifies the properties of the model due to the multiplicativity of the noise.…”

Section: Introductionmentioning

confidence: 99%

The Jacobi diffusion process as a neuronal model

D’Onofrio

Tamborrino

Chaos: An Interdisciplinary Journal of Nonlinear Science

2018

Self Cite

The Jacobi process is a stochastic diffusion characterized by a linear drift and a special form of multiplicative noise which keeps the process confined between two boundaries. One example of such a process can be obtained as the diffusion limit of the Stein’s model of membrane depolarization which includes both excitatory and inhibitory reversal potentials. The reversal potentials create the two boundaries between which the process is confined. Solving the first-passage-time problem for the Jacobi process, we found closed-form expressions for mean, variance, and third moment that are easy to implement numerically. The first two moments are used here to determine the role played by the parameters of the neuronal model; namely, the effect of multiplicative noise on the output of the Jacobi neuronal model with input-dependent parameters is examined in detail and compared with the properties of the generic Jacobi diffusion. It appears that the dependence of the model parameters on the rate of inhibition turns out to be of primary importance to observe a change in the slope of the response curves. This dependence also affects the variability of the output as reflected by the coefficient of variation. It often takes values larger than one, and it is not always a monotonic function in dependency on the rate of excitation.