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Ultimo aggiornamento	12 set 2013
Autore	Simona Olmi
Sesso	M
Esperimento	PIECES
Tipo	Dottorato
Destinazione dopo il cons. del titolo	Post-dottorato (Italia)
Università	Università Di Firenze
Strutt.INFN/Ente	Firenze
Titolo	Collective Dynamics in Complex Neural Networks
Abstract	In this dissertation we used simple mathematical models to describe neural network dynamics and to analyze the emergence of collective behaviour in these networks. In particular we investigated the role played by the topology in promoting coherent activity in excitatory pulse-coupled networks. The presence of collective oscillations assume greater importance in neuroscience since coherent oscillatory activities are prominent in the cortex of the awake brain during attention and are involved in higher level processes, such as sensory binding, storage of memories, and even consciousness. The origin of these oscillations has been commonly associated with the balance between excitation and inhibition in the network, while purely excitatory circuits are believed to lead only to fully synchronized regimes, therefore to a quite trivial macroscopic dynamics. Furthermore in neuroscience a completely synchronized firing is a symptom of neural disorders, like e.g. Parkinson disease. However, recent “ex vivo” measurements performed on the rodent neocortex and hippocampus in the early stage of brain maturation reveal coherent activity patterns, despite the fact that the synapses are completely excitatory at this stage while inhibitory synapses develop only later. For the above reasons, numerical and analytical studies of collective motions in networks of simple spiking neurons have been mainly devoted to balanced excitatory-inhibitory configurations, while few studies focused on the emergence of coherent activity in purely excitatory networks. In particular pioneristic studies on coherent periodic activity in fully coupled excitatory networks of leaky integrate-and-fire neurons revealed a regime characterized by a partial synchronization at the population level, while the single neurons perform quasi-periodic motions. Since in the neural cortex the presence of noise is an intrinsic property and real neural circuits are not expected to have a full connectivity, it is important to investigate whether the partially synchronized regime will survive to strong level of dilution and to the introduction of different topologies. On the other hand it has been already shown that the partially synchronized regime is quite robust to moderate levels of noise. Therefore we analyzed the influence on the dynamics of these models of different topologies. In particular we considered an excitatory random network where neurons are connected as in a directed Erdoes- Renyi graph with average connectivity scaling sublinearly with the number of neurons in the network. In these “massively connected” networks the dynamics of coherent collective states is similar to the one found in globally coupled networks. However, due to the disorder present in the system, for finite number of neurons we have inhomogeneities in the neuronal behaviors,inducing a weak form of chaos, which vanishes in the thermodynamic limit. In this limit, the disordered systems exhibit regular (non chaotic) dynamics thus recovering the properties of a homogeneous fully connected network. Similar results will be expected in different dynamical systems, once considered a massively coupled network. On the other hand, the situation is completely different for a “sparse network” characterized by a constant connectivity, independent on the size of the network. In fact, by increasing the connectivity value, we found that a few tens of random connections are sufficient to sustain a nontrivial (and possibly irregular) collective dynamics. In other words, collective motion is a rather generic and robust property and does not require an extremely high connectivity to be sustained. Moreover, the collective motion coexists with a microscopically chaotic dynamics that does not vanish in the thermodynamic limit and turns out to be extensive (the number of unstable directions is proportional to the network size). More specifically, we studied various classes of dynamical models on random sparse networks and in all cases, irrespective of the presence of the macroscopic phase, we found that the chaotic dynamics is always extensive. Extensive chaos has been already found in spatially extended system with nearest-neighbour coupling (diffusive coupling) induced by the additivity of the system. In our case this property is highly nontrivial, as the network dynamics is non additive and it cannot be approximated as the juxtaposition of almost independent sub-structures. This is one of the fundamental results found in this dissertation. Furthermore we studied the dynamics of two symmetrically coupled populations of leaky integrate- and-fire neurons characterized by excitatory coupling: this is the simplest instance of “network-of- networks”, that is often invoked as a paradigm for neural system. Even though the system is fully coupled, an inhomogeneity is present due to the fact that self-coupling among the neurons in the same population is different from the cross-coupling among the neurons in the other population. Upon varying both the self-coupling and the cross-coupling strengths we found various kinds of symmetric and symmetry broken collective states some of which are similar to states previously observed, while other are completely new. In addition to this,since single neurons are symmetrically coupled, the symmetry breaking is completely spontaneous. This setup has revealed, for the first time in neural networks, the onset of “chimera states”, where one of the two populations is fully synchronized, while the oscillators of the other one are not synchronized at all. So far this kind of symmetry broken states has been observed only in oscillators’ networks, while the existence of these states in neural networks have been just conjectured. Furthermore we have found macroscopic states of increasing complexity, characterized by collective periodic oscillations, quasi-periodic motion or even macroscopic chaos. Collective chaos has been observed so far in coupled map lattices or in oscillator networks but never in neural networks. The collective origin of the chaotic motion is revealed by the Finite Size Lyapunov Exponent analysis and confirmed by comparing the previous results with the standard Lyapunov analysis. In addition to the numerical analysis previously described, we carried out a broad and prime analytical study about the stability of a largely investigated collective state,the so called “splay state” (or asynchronous state), in networks of N globally pulse- coupled phase-like models of neurons. In a splay state, all neurons follow the same periodic dynamics and their phases are evenly shifted: they fire periodically with period T and two successive spike emissions occur at regular intervals T/N. Our model neurons are characterized by a membrane potential u that is continuously driven by the velocity field F(u), from the resetting value u = 0 towards the threshold u = 1. As threshold and resetting value can be identified with one another and thereby u interpreted as a phase, it will be customary to refer to the case F(1) 6= F(0) as to that of a discontinuous velocity field. For the simple choice F(u) = a?u, the model reduces to the well known case of LIF neurons. In this case we considered both excitatory and inhibitory networks. The study of the stability requires determining the Floquet spectrum, i.e. the complex eigenvalues associated to a given periodic orbit of period T. By developing a perturbative approach that is valid for arbitrary coupling strength we were also able to determine the spectral shape and find it to be independent of the structure of the velocity field. In the large-N limit it is natural to consider 1/N as a smallness parameter and thereby to expand the evolution equations in powers of 1/N since two successive spikes are emitted in an interval T/N. Furthermore we proved that, in the thermodynamic limit, the Floquet spectrum scales as (1/N2) for generic discontinuous velocity fields. For continuous fields, it has been numerically observed that the scaling of the spectrum is at least O(1/N^4). In other words the shape of the spectrum is universal, apart from a multiplicative factor that vanishes if and only if F(1) = F(0), i.e. for true phase rotators where u = 0 coincides with u = 1. The stability of the splay state can be inferred, for arbitrary coupling strength, from the sign of F(1) ? F(0): in excitatory (inhibitory) networks, the state is stable whenever F(0) > F(1) (F(0) < F(1)). Finally, another obtained analytical result regards the dependence of the period T onto the size N which is of order o(1/N^3).
Anno iscrizione	2010
Data conseguimento	22 mar 2013
Luogo conseguimento	Firenze
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