Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function

Giuseppe Chindemi, Pawel Tadeusz Dlotko, Kathryn Hess Bellwald, Ran Levi, Henry Markram, Max Christian Nolte, Rodrigo de Campos Perin, Michael Reimann, Martina Scolamiero, Katharine Felicity Turner
Frontiers Media Sa, 2017
Article

Résumé

The lack of a formal link between neural network structure and its emergent function has hampered our understanding of how the brain processes information. We have now come closer to describing such a link by taking the direction of synaptic transmission into account, constructing graphs of a network that reflect the direction of information flow, and analyzing these directed graphs using algebraic topology. Applying this approach to a local network of neurons in the neocortex revealed a remarkably intricate and previously unseen topology of synaptic connectivity. The synaptic network contains an abundance of cliques of neurons bound into cavities that guide the emergence of correlated activity. In response to stimuli, correlated activity binds synaptically connected neurons into functional cliques and cavities that evolve in a stereotypical sequence toward peak complexity. We propose that the brain processes stimuli by forming increasingly complex functional cliques and cavities.

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Publications associées (32)

Sensory memory in a model of a cortical column

Julien Mayor

Sensory memory is the first step of a complex memorization process. Sensory information is buffered in a "sensory store" for a short period. Then part of it is transferred to working memory, where information can be actively processed and, eventually, through its rehearsal, can be memorized and stored in long-term memory. Sensory memory is characterized by a high capacity and a short temporal extension. In this work we show that such a transient buffering of information is well captured in models of cortical microcircuits. Sensory information stays for a short time in the perturbations it induced to the neural activity. Two fundamental characteristics of cortical columns appear to be useful to this purpose. The first one is an approximate balance of neuronal excitation and inhibition. The second one is the sparseness of cells recurrent connections. Together, these two discriminants ensure a broad set of complex dynamical behaviours of the neural network along with the presence of a phase of sustained non-oscillatory activity with very irregular spike trains in individual neurons. We show that at the vicinity of a phase transition from a quiescent state to a phase of chaotic spiking activity, global stimuli can be buffered in the "macroscopic" neuronal activity. However if the sensory input only excites a subpart of the network, efficient information buffering is made using the detailed spiking activity of every neuron in the network. The "microscopic" structure of the network is therefore needed in order to buffer sensory information. In most situations, addition of noise enhances the buffering performance and the computational power of the network. This effect is known in the physical literature as "stochastic resonance". If the network has to achieve complex transformation of its inputs, stochastic resonance is shown to be an emergent property of the population of neurons. Both the very irregular spiking activity present in vivo and the balanced excitation and inhibition measured in cortical columns have a new functional relevance in being the neuronal substrate of an efficient sensory memory .

EPFL2005

Chargement

Coalgèbres d'Alexander-Whitney

Théophile Naïto

The goal of this work is to study Alexander-Whitney coalgebras (first defined in [HPST06]) from a topological point of view. An Alexander-Whitney coalgebra is a coassociative chain coalgebra over Z with an extra algebraic structure : the comultiplication must respect the coalgebra structure up to an infinite sequence of homotopies (this sequence is part of the data of the Alexander-Whitney coalgebra structure). Alexander-Whitney coalgebras are interesting for topologists because the normalized chain complex C(K) of a simplicial set K is endowed with an Alexander-Whitney coalgebra structure. This theorem is proved for the first time here (generalising a result proven in [HPST06]). This theorem gives the hope that the Alexander-Whitney coalgebra structure of C(K) contains interesting information that can be used to solve topological problems. This hope is strengthened by the success already obtained in the work of several topologists. Among others, [HPST06], [HL07], [Boy08], and [HR] use the Alexander-Whitney coalgebra structure of the normalized chains of a simplicial set in an essential way to solve topological problems. This thesis begins with some background material. In particular, the definition of a DCSH morphism between two coassociative chain coalgebras is recalled in complete detail. For example, signs are determined with great precision. Next we devote a chapter to the definition of Alexander-Whitney coalgebras and to their importance in topology. In the following chapter we begin the conceptual study of Alexander-Whitney coalgebras. A global study of these objects had not yet been carried out even if the Alexander-Whitney coalgebra structure has been studied and used in order to answer some specific questions. With the aim of studying Alexander-Whitney coalgebras in a nice setting, we develop an operadic description of these coalgebras in the following chapter. More precisely, we show that there is an explicit operad AW such that the coalgebras over this operad are exactly the Alexander-Whitney coalgebras. Furthermore, AW is shown to be a Hopf operad, so that the category formed by the Alexander-Whitney coalgebras is actually a monoidal category. These results are proven in a reasonably general framework. In fact, we associate an operad to each bimodule (over the associative operad) of a certain type, such that we get AW if this bimodule is well chosen. In particular, these results enable us to study Alexander-Whitney coalgebras from the standpoint of operads. This strategy is recognised to be successful in various mathematical situations, and especially in algebraic topology. Moreover, we develop a minimal model notion in the setting of right module over a chosen operad (which has to satisfy some reasonable conditions), with the aim of applying this result to the special case of the Alexander-Whitney coalgebras. This is possible because coalgebras over some fixed operad P can be seen as right modules over P. And the category of right modules over P has some nice features which do not appear to hold in the category of P-coalgebras. The inspiration for this part of our work comes from the notion of minimal model developed in the framework of rational homotopy theory. The two following facts show that it is reasonable to try to adapt some ideas of rational homotopy theory to the category of Alexander-Whitney coalgebras. A. There is a theorem that says that studying topological spaces up to rational equivalences is, essentially, equivalent to studying cocommutative chain coalgebras over the field of rational numbers. This is false if the ring of integers replaces the field of rational numbers, but Alexander-Whitney coalgebras are "almost" cocommutative in the sense which is explained in this thesis. B. It could be that the Alexander-Whitney coalgebra structure of the normalized chains of a simplicial set is weak enough to allow explicit computations. At least, it is clear that the Alexander-Whitney coalgebra structure on the normalized chains is far from being an E∞-structure (such a structure determines the homotopy type of the considered simplicial set, at least under some conditions). The chapter about minimal models in the framework of right modules over an operad includes an existence theorem and a discussion of the unicity of this model. In the second part of this chapter, we construct an explicit path-object in the model category of right modules over an operad. This path-object is then used to investigate the topologically relevant information that could stem from the minimal model in the case of the operad AW. Finally, we present and examine some interesting open questions about Alexander-Whitney coalgebras. These questions give a nice outlook on future research in this area.

EPFL2009

Chargement

Since the seminal work of Watts in the late 90s [1], graph-theoretic analyses have been performed on many complex dynamic networks, including brain structures [2]. Most studies have focused on functional connectivity defined between whole brain regions, using imaging techniques such as fMRI, EEG or MEG. Only very few studies have attempted to look at the structure of neural networks on the level of individual neurons [3,4]. To the best of our knowledge, these studies have only considered undirected connectivity networks and have derived connectivity based on estimates on small subsets or even pairs of neurons from the recorded networks.Here, we investigate scale-free and small-world properties of neuronal networks, based on multi-electrode recordings from the awake monkey on a larger data set than in previous approaches. We estimate effective, i.e. causal, interactions by fitting Generalized Linear Models on the neural responses to natural stimulation. The resulting connectivity matrix is directed and a link between two neurons represents a causal influence from one neuron to the other, given the observation of all other neurons from the population. We use this connectivity matrix to estimate scale-free and small-world properties of the network samples. For this, the quantity proposed by Humphries et al. (2008) for quantifying small-world-ness is generalized to directed networks [5]. We find that the networks under consideration lack scale-free behavior, but show a small, but significant small-world structure.Finally, we show that the experimental design of multi-electrode recordings typically enforces a particular network structure that can have a considerate impact on how the small-world structure of the network should be evaluated. Random graphs that take the geometry of the experiment into account can serve as a more refined null model than the homogeneous random graphs that are usually proposed as reference models to evaluate small-world properties. References [1] Watts, D. J., Strogatz, S. H., 1998. Collective dynamics of 'small-world' networks. Nature 393 (6684), 440-442. [2] Bullmore, E., & Sporns, O. (2009). Complex brain networks: graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience, 10(3), 186–198. [3] Bettencourt, L. M. A., Stephens, G. J., Ham, M. I., Gross, G. W., 2007. Functional structure of cortical neuronal networks grown in vitro. Physical Review E 75 (2), 021915+. [4] Yu, S., Huang, D., Singer, W., Nikolic, D., 2008. A small world of neuronal synchrony. Cereb. Cortex, bhn047+. [5] Humphries, M. D., Gurney, K., April 2008. Network 'small-world-ness': a quantitative method for determining canonical network equivalence. PloS one 3 (4), e0002051+.

2010

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Sensory memory in a model of a cortical column

Julien Mayor

EPFL2005

Coalgèbres d'Alexander-Whitney

Théophile Naïto

EPFL2009

2010