Linear Response of General Observables in Spiking Neuronal Network Models

Cessac, Bruno; Ampuero, Ignacio; Cofré, Rodrigo

doi:10.3390/e23020155

Cited by 8 publications

(21 citation statements)

References 66 publications

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“…The second model, introduced in Section 2.2 and analysed in Section 3.3, attempts to propose a canonical form of probability distributions of the retinal spike trains based on the constraints (i), (ii), (iii) above. These sections essentially present the conclusions of works published elsewhere [46][47][48][49][50][51]. As I argue, these constraints lead to a natural notion of spike probabilities, somewhat extending the statistical physics notion of Gibbs distribution to the non-stationary case.…”

Section: Introductionmentioning

confidence: 82%

“…It is inherited from statistical physics on one hand [75] and on extensions of Markov chains to unbounded memory on the other hand [76]. The material briefly sketched here has been published in [46][47][48][49][50][51].…”

Section: Spike Statisticsmentioning

confidence: 99%

“…Note that the integration of ( 27) does not only require the knowledge of voltages V k , stimulus and noise at time t. It requires, in addition, the knowledge of the spike train ω emitted by the network before t. In this sense, this is not a classical dynamical system. Nevertheless, Equation ( 27) can be explicitly integrated [47,51].…”

Section: A Model Of Effective Interactions Between Rg Cellsmentioning

confidence: 99%

“…The model here is a classical leaky integrate and fire model in the presence of noise, where parameters have been tuned to have a spontaneous asynchronous activity as depicted in Figure 3 (bottom,left). See [51] for more details. Consider a moving stimulus S(x, t) propagating from left to right (cyan, bell shaped curve) Figure 3 (top).…”

Section: Conclusion Of This Sectionmentioning

confidence: 99%

“…Indeed, spike correlations are modified by a spatio-temporal stimulus and can be computed thanks to the knowledge of spontaneous correlations. The linear response formula is expressed as a convolution where the kernel can be explicitly computed for an Integrate and Fire conductance based model [51]. Moreover, as I argue, these spike train distributions have close links with information geometry.…”

Section: Introductionmentioning

confidence: 99%

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Retinal Processing: Insights from Mathematical Modelling

Cessac

2022

J. Imaging

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The retina is the entrance of the visual system. Although based on common biophysical principles, the dynamics of retinal neurons are quite different from their cortical counterparts, raising interesting problems for modellers. In this paper, I address some mathematically stated questions in this spirit, discussing, in particular: (1) How could lateral amacrine cell connectivity shape the spatio-temporal spike response of retinal ganglion cells? (2) How could spatio-temporal stimuli correlations and retinal network dynamics shape the spike train correlations at the output of the retina? These questions are addressed, first, introducing a mathematically tractable model of the layered retina, integrating amacrine cells’ lateral connectivity and piecewise linear rectification, allowing for computing the retinal ganglion cells receptive field together with the voltage and spike correlations of retinal ganglion cells resulting from the amacrine cells networks. Then, I review some recent results showing how the concept of spatio-temporal Gibbs distributions and linear response theory can be used to characterize the collective spike response to a spatio-temporal stimulus of a set of retinal ganglion cells, coupled via effective interactions corresponding to the amacrine cells network. On these bases, I briefly discuss several potential consequences of these results at the cortical level.

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Section: Introductionmentioning

confidence: 82%

Section: Spike Statisticsmentioning

confidence: 99%

Section: A Model Of Effective Interactions Between Rg Cellsmentioning

confidence: 99%

Section: Conclusion Of This Sectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Retinal Processing: Insights from Mathematical Modelling

Cessac

2022

J. Imaging

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Fluctuation–response relations for integrate-and-fire models with an absolute refractory period

Puttkammer,

Lindner

2024

Biol Cybern

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We study the problem of relating the spontaneous fluctuations of a stochastic integrate-and-fire (IF) model to the response of the instantaneous firing rate to time-dependent stimulation if the IF model is endowed with a non-vanishing refractory period and a finite (stereotypical) spike shape. This seemingly harmless addition to the model is shown to complicate the analysis put forward by Lindner Phys. Rev. Lett. (2022), i.e., the incorporation of the reset into the model equation, the Rice-like averaging of the stochastic differential equation, and the application of the Furutsu–Novikov theorem. We derive a still exact (although more complicated) fluctuation–response relation (FRR) for an IF model with refractory state and a white Gaussian background noise. We also briefly discuss an approximation for the case of a colored Gaussian noise and conclude with a summary and outlook on open problems.

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Response theory identifies reaction coordinates and explains critical phenomena in noisy interacting systems

Zagli,

Lucarini,

Pavliotis

2024

J. Phys. A: Math. Theor.

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We consider a class of nonequilibrium systems of interacting agents with pairwise interactions and quenched disorder in the dynamics featuring, in the thermodynamic limit, phase transitions. We identify mathematical conditions on the microscopic interaction structure, namely the separability of the interaction kernel, that lead to a dimension reduction of the system in terms of a finite number of reaction coordinates. Such reaction coordinates prove to be proper nonequilibrium thermodynamic variables as they carry information on correlation, memory and resilience properties of the system. Phase transitions can be identified and quantitatively characterised as singularities of the complex valued susceptibility functions associated to the reaction coordinates. We provide analytical and numerical evidence of how the singularities affect the physical properties of finite size systems.

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Linear Response of General Observables in Spiking Neuronal Network Models

Cited by 8 publications

References 66 publications

Retinal Processing: Insights from Mathematical Modelling

Retinal Processing: Insights from Mathematical Modelling

Fluctuation–response relations for integrate-and-fire models with an absolute refractory period

Response theory identifies reaction coordinates and explains critical phenomena in noisy interacting systems

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