Speaker: Emre Baspinar (Paris-Saclay Institute of Neuroscience (Neuro-PSI), CNRS (Campus CEA, Saclay), Laboratory of Computational Neuroscience).

Summary:

We will divide the mini course into three parts of which each one is dedicated to a different modelling approach. Each approach is applied to one of the following neuroscientific problems: decision making, neuronal dynamics, visual perception. The first approach is based on a biophysically plausible mean field model extended from an excitatory-inhibitory population mean field model of adaptive exponential integrate-and-fire setting [1]. This extension is a generic multipopulation and multipool mean field framework which can be employed for many neuroscientific problems, in particular for decision making as we will see a specific example of.

The second approach is based on multiple time scale stochastic dynamical systems at both network and mean field levels. We will focus on a stochastic Wilson-Cowan type elliptic bursting network system and its mean field framework. The network produces particular degenerate dynamical patterns in the transition regime between elliptic bursting and tonic spiking. Those degenerate patterns are of particular interest since they mark the beginning and ending of the transition, thus possibly changes in the neuronal activity regimes. Those transitive patterns are called torus canards. They can be captured and classified by studying a canonical model attributed to the mean field setting derived from the Wilson-Cowan type network. We will present the network, show a new family of torus canards at network level and then provide the canonical model as well as its singular limit which allows us to capture and classify the new family of torus canards [2].

The third and the final approach belongs to the so-called neurogeometric method [3, 4, 5]. We will present a geometric model for the orientation, spatial frequency and phase selective behavior of the primary visual cortex [6]. The model framework is based on a sub-Riemannian geometry which is derived from one of the very first mechanisms of the mam- malian visual perception: receptive profile. We will use this model geometry to represent the orientation, spatial frequency and phase values extracted by the receptive profiles from an input visual stimulus. Furthermore, we will show that the long range cortical connectivity can be modelled in terms of a family of integral curves within this geometry. We will see the applications of this model geometry to generating multiscale orientation prefence maps found in the visual cortex [7], and to reproducing Poggendorff-type visual illusions [8]. Finally, we will mention some potential links of those applications to pathological visual perception in certain types of migraine and epilepsy.

References:

[1]  M. Di Volo, A. Romagnoni, C. Capone, and A. Destexhe, “Biologically re- alistic mean-field models of conductance-based networks of spiking neurons with adaptation,” Neural Computation, vol. 31, no. 4, pp. 653–680, 2019.

[2]  E. Baspinar, D. Avitabile, and M. Desroches, “Canonical models for torus canards in elliptic bursters,” Chaos: An Interdisciplinary Journal of Non- linear Science, vol. 31, no. 6, p. 063129, 2021.

[3]  W. C. Hoffman, “The visual cortex is a contact bundle,” Applied Mathemat- ics and Computation, vol. 32, no. 2-3, pp. 137–167, 1989.

[4]  J. Petitot and Y. Tondut, “Vers une neurog ́eom ́etrie: fibrations corticales, structures de contact et contours subjectifs modaux,” Math ́ematiques et Sci- ences Humaines, vol. 145, pp. 5–101, 1999.

[5]  G. Citti and A. Sarti, “A cortical based model of perceptual completion in the roto-translation space,” Journal of Mathematical Imaging and Vision, vol. 24, no. 3, pp. 307–326, 2006.

[6]  E. Baspinar, A. Sarti, and G. Citti, “A sub-Riemannian model of the vi- sual cortex with frequency and phase,” The Journal of Mathematical Neu- roscience, vol. 10, no. 1, pp. 1–31, 2020.

[7]  E. Baspinar, G. Citti, and A. Sarti, “A geometric model of multi-scale ori- entation preference maps via Gabor functions,” Journal of Mathematical Imaging and Vision, vol. 60, no. 6, pp. 900–912, 2018.

[8]  E. Baspinar, L. Calatroni, V. Franceschi, and D. Prandi, “A cortical-inspired sub-Riemannian model for poggendorff-type visual illusions,” Journal of Imaging, vol. 7, no. 3, p. 41, 2021.