# Invited Lectures

Pascal Chossat (University Nice Sophia Antipolis, France)

Title : On the effects of the pinwheel network symmetries on cortical response

Abstract :

A recent work is presented, in which the mathematical description of the spontaneous activity of V1 was revisited by combining several important experimental observations including 1) the organization of the visual cortex into a spatially periodic network of hypercolumns structured around pinwheels, 2) the difference between short-range and long-range intra-cortical connections, the first ones being rather isotropic and producing naturally doubly-periodic patterns by Turing mechanisms, the second one being patchy and 3) the fact that the Turing patterns spontaneously produced by the short-range connections and the network of pinwheels have similar periods.

By analyzing the Preferred Orientation (PO) map, we are able to classify all possible singular points of the PO maps (the pinwheels) as having symmetries described by a small subset of the wallpaper groups. We then propose a description of the spontaneous activity of V1 using a classical voltage-based neural field model that features isotropic short-range connectivities modulated by non-isotropic long-range connectivities. A key observation is that, with only short-range connections and because the problem has full translational invariance in this case, a spontaneous doubly-periodic pattern generates a 2-torus in a suitable functional space which persists as a flow-invariant manifold under small perturbations, hence when turning on the long-range connections. Through a complete analysis of the symmetries of the resulting neural field equation and motivated by a numerical investigation of the bifurcations of their solutions, we conclude that the branches of solutions which are stable over an extended set of parameters are those corresponding to patterns with an hexagonal (or nearly hexagonal) symmetry.

The question of which patterns persist when turning on the long-range connections is answered by 1) analyzing the remaining symmetries on the perturbed torus and 2) combining this information with the Poincar´e-Hopf theorem to select the possible generic patterns and dynamics that can occur. We have developed a numerical implementation of the theory that has allowed us to produce the patterns of activities predicted by the theory. In particular we generalize the contoured and non-contoured planforms predicted by previous authors [1, 2] and predict the existence of mixed contoured/non-contoured planforms. We also found that these planforms are most likely to be time dependent.

References

[1] GB Ermentrout and JD Cowan. A mathematical theory of visual hallucination patterns, Biological cybernetics 1 (2) pp. 1234-1245, 1979.

[2] P.C. Bressloff, J.D. Cowan, M. Golubitsky, P.J. Thomas, and M.C. Wiener. Geo- metric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex, Phil. Trans. R. Soc. Lond. B 306(1407):299330, 2001.

Jean-Pierre Francoise (Université Pierre et Marie Curie, France)

Title: New directions in bifurcation theory emerging from modeling neurosciences

Abstract :

Several useful models developed in neurosciences, in particular within the weakly coupled oscillators approach, use bifurcation theory and fast-slow systems. This talk will present complex analysis and singularity theory methods to investigate some mathematical problems of this field, including fast-slow version of the Bautin bifurcation, path of periodic orbits connecting two bifurcations, and deformations of bursting oscillations.

Maciej Krupa (Inria, France)

joint work with

Mathieu Desroches (Inria, France)

Tasso Kaper (Boston University, USA)

Title: The role of canards in transition to bursting and spike adding

Abstract :

Bursting is one of the essential behaviors observed in cellular activity, in relation to either electrical or secretion events. The main characteristic of a bursting signal is the alternation between slow (silent) phases with no oscillations or small-amplitude (sub threshold) oscillations, and fast (active) phases with large-amplitude oscillations (groups of spikes). Since the pioneering work of Rinzel, bursting has been extensively analyzed in ODE models that possess multiple timescales. The underlying slow-fast structure of these models allows to understand the overall features of the bursting solution as a succession of slow passages near families of attractors of the fast subsystem interspersed with periods of slow evolution. Rinzel proposed a classification of bursting patterns into three main classes: square-wave, elliptic and parabolic. We revisit these three bursting classes and explain how, in every case, the transition from oscillatory dynamics to bursting is organized by so-called canard solutions, which also turn out to play an essential role in controlling the number of spikes per burst upon variation of a control parameter. We distinguish three different canard based bursting mechanisms: transition to square wave bursting through a generalized canard explosion, transition from fast oscillation to elliptic bursting through torus canards and transition from slow oscillation to parabolic bursting through a homoclinic-like bifurcation involving a folded saddle and the associated canard.

Joscha Liedtke, MPI for Dynamics and Self-Organization, Germany

joint work with

Fred Wolf, MPI for Dynamics and Self-Organization, Germany

Title : Genetic networks specifying the functional architecture of orientation domains in V1

Abstract :

Although genetic information is critically important for brain development and structure, it is widely believed that neocortical functional architecture is largely shaped by activity dependent mechanisms. The information capacity of the genome simply appears way too small to contain a blueprint for hardwiring the cortex. Here we show theoretically that genetic mechanisms can in principle circumvent this information bottleneck. We find in mathematical models of genetic networks of principal neurons interacting by long range axonal morphogen transport that morphogen patterns can be generated that exactly prescribe the functional architecture of the primary visual cortex (V1) as experimentally observed in primates and carnivores. We analyze in detail an example genetic network that encodes the functional architecture of V1 by a dynamically generated morphogen pattern. We use analytical methods from weakly non-linear analysis[1] complemented by numerical simulation to obtain solutions of the model. In particular we find that the pinwheel density variations, pinwheel nearest neighbor distances and most strikingly the pinwheel densities are in quantitative agreement with high precision experimental measurements[2]. We point out that the intriguing hypothesis that genetic circuits coupled through axonal transport shape the complex architecture of V1 is in line with several biological findings. (1) Surprisingly, transcription factors have been found to be transported via axons and to be incorporated in the nucleus of the target cells[3]. (2) A molecular correlate was recently found for ocular dominance columns in V1[4]. (3) We estimate that the speed of axonal transport is rapid enough to achieve appropriate timescales. This theory opens a novel perspective on the experimentally observed robustness of V1’s architecture against radically abnormal developmental conditions such a dark rearing[5]. Furthermore, it provides for the first time a scheme how the pattern of a complex cortical architecture can be specified using only a small genetic bandwidth.

References

[1] Cross, M., & Hohenberg, P. (1993). Reviews of Modern Physics

[2] Kaschube, M. et al. (2010). Science

[3] Sugiyama, S. et al. (2008). Cell

[4] Tomita, K. et al. (2012). Cerebral Cortex

[5] White, L. E. et al. (2001). Nature

Andre Longtin (University of Ottawa, Canada)

Title: Paradoxical oscillations from feedforward networks

Abstract:

Oscillations in the activity of neural networks are generally associated with recurrent circuitry, a notion that has guided the search for important mechanisms that lead to e.g. gamma rhythm generation in response to correlated inputs. Here we find that under fairly general conditions, feedforward circuitry can mimic oscillatory behaviour, yielding similar first and second-order spike train statistics. This leads to novel hypotheses for the occurrence and prevalence of such rhythms. The numerical simulations are supported by linear response theory which is built on the frequency response of stochastic leaky integrate-and-fire neurons. The results also highlight the partial equivalence of feedforward and feedback circuitry.

Alessandro Sarti, CAMS, CNRS-EHESS, France

joint work with

Giovanna Citti, Mathematics Department, Universita di Bologna, Italy

Title : A Gauge theory for coupling cortical layers

Wilhelm Stannat (Technische Universität Berlin, Germany)

Title: Stochastic neural field equations

Abstract :

Neural field equations are used to describe the spatiotemporal evolution of the average activity in a network of synaptically coupled populations of neurons in the continuum limit. This deterministic description is only accurate in the infinite population limit and the actual finite size of the populations causes deviations from the mean field behavior. One way to take into account fluctuations is to add noise to the neural field equations. The resulting class of stochastic neural field equations exhibit a rich phenomenology and it is the main aim of this talk to introduce a complete mathematical framework for the rigorous multiscale analysis of stochastic neural field equations.

Stochastic neural field equations have first been introduced in the paper [2] to study various effects of multiplicative noise on traveling waves. First rigorous results concerning well-posedness of these equations appeared in the paper [3], while dynamical equations for the asymptotic fluctuations in approximating Markov chains have formally been identified in [6].

In our talk we will first rigorously derive stochastic neural field equations with noise terms accounting for finite size effects. These equations are identified by describing the evolution of the activity of finite-size populations by Markov chains and then determining their asymptotic fluctuations. The jump rates are of a different form than considered in the literature so far (see [1]) and lead to qualitatively different results. In particular the fluctuations around stable stationary solutions are of smaller order than previously assumed. We then introduce a complete mathematical framework for the analysis of the resulting stochastic neural field equations.

As first steps of a multiscale analysis, a geometrically motivated decomposition of the stochastic evolution into a randomly moving wave front and fluctuations is derived next. A random ordinary differential equation describing the velocity of the moving wave front can be deduced and the fluctuations around the wave front turn out to be non-Gaussian, even if the driving noise term is a Gaussian process. We will compare our results with the findings on the impact of noise on the position and velocity of the wave front obtained in [2]. Thereby we will summarize recent results on the Lyapunov stability of wave fronts, necessary for our analysis.

The presented geometric approach is in principle applicable to describe the statistics of any macroscopic profile driven by spatially extended noise, like, e.g., wave fronts and pulses in general stochastic reaction diffusion systems

(see [7]).

The talk is partially based on joint work with Jennifer Krüger ([4]) and Eva Lang ([5]).

References

[1] P.C. Bressloff, Stochastic neural field theory and the system-size expansion, SIAM Journal on Applied Mathematics, 70:14881521, 2009.

[2] P.C. Bressloff, M.A. Webber, Front propagation in stochastic neural fields, SIAM J. Appl. Dyn. Syst., 11:708740, 2012.

[3] O. Faugeras, J. Inglis, Stochastic neural field equations: A rigorous footing, J. Math. Biol., DOI 10.1007/s00285-014-0807-6, 2014.

[4] J. Krüger, W. Stannat, Front Propagation in Stochastic Neural Fields: A rigorous mathematical framework, SIAM J. Appl. Dyn. Syst., 13:1293-1310, 2014.

[5] E. Lang, W. Stannat, Finite-Size effects on traveling wave solutions to neural field equations, submitted, 2014.

[6] M.G. Riedler, E. Buckwar, Laws of large numbers and Langevin approximations for stochastic neural field equations, J. Math. Neurosci., 3, 2013.

[7] W. Stannat, Stability of travelling waves in stochastic bistable reaction-diffusion equations, arXiv:1404.3853

Michèle Thieullen (Université Pierre et Marie Curie, France)

joint work with

R. Höpfner, (J. Gutenberg-Univ. Mainz, Germany)

E. Löcherbach, (Université de Cergy-Pontoise, France)

Title : A stochastic Hodgkin-Huxley model with periodic input

Abstract :

We consider an extended stochastic Hodgkin-Huxley model for a spiking neuron including its dendritic input. The latter carries some deterministic periodic signal coded in its drift coefficient and is the only source of noise of the whole system. This amounts to a 5d SDE driven by 1d Brownian motion for which we can prove positive Harris recurrence.

This approach provides us with laws of large numbers which allow to describe the spiking activity of the neuron in the long run. This talk is based on our recent work [1], [2], [3].

References

[1] Höpfner, R., Löcherbach, E., Thieullen, M. Strongly degenerate time inhomogenous SDEs: densities and support properties. Application to a Hodgkin-Huxley system with periodic input, Preprint arXiv:1410.0341

[2] Höpfner, R., Löcherbach, E., Thieullen, M. Ergodicity for a stochastic Hodgkin-Huxley model driven by Ornstein-Uhlenbeck type input*, to appear in* A. I. H. Poincaré, arXiv:1311.3458v3

[3] Höpfner, R., Löcherbach, E., Thieullen, M. Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Application to a stochastic Hodgkin-Huxley model, In preparation.

Stephan van-Gils (University of Twente, the Netherlands)

Title: Using sun-star calculus for normal form calculations in neural field equations

Abstract:

Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we present an extensive study of two pitchfork-Hopf bifurcations for a 1D neural field model with ‘Wizard hat’ type connectivity.

- van Gils, S. A., Janssens, S. G., Kuznetsov, Y. A., & Visser, S. (2013). On local bifurcations in neural field models with transmission delays.
*Journal of mathematical biology*,*66*(4-5), 837-887. - Dijkstra, K., van Gils, S. A., Janssens, S. G., Kuznetsov, Y. A., & Visser, S. (2015). Pitchfork-Hopf bifurcations in 1D neural field models with transmission delays.
*Physica D: Nonlinear Phenomena*.

Martin Wechselberger (University of Sidney, Australia)

Title: The role of cell volume changes in normal and pathological dynamics of the brain

Abstract:

I will discuss a recent study of the effect of cell volume on neural dynamics by incorporating cell volume changes together with dynamic ion concentrations and oxygen supply into Hodgkin-Huxley spiking dynamics. I will highlight to role of multiple time scale structure in these extended conductance based models and show how transitions from normal to pathological states arise.