In last years, also thanks to the development of more and more powerful calculators, the scientific community has begun to leave the study of simple networks --- e.g. Hopfield networks where neurons are modeled as binary variables --- to more elaborated networks where the dynamics of the membrane potential of each neuron is described by continuous variables evolving according to one or more differential equations. With the help of physiological measurements, networks with more realistic couplings --- where the information is transmitted via pulses of finite duration --- have been developed. These pulse coupled networks have been widely employed both for computational neuroscience studies as well as images processing.
More in details, in this project we consider networks where neurons interact via finite pulses transmission. The interest in this kind of models is due to theoretical analysis but also to experimental evidences. Previous analytical and numerical studies of globally coupled networks with excitatory synapses show how in these systems stable collective states --- which are characterized by nontrivial dynamics --- can spontaneously emerge. These "Partially Synchronized" (PS) states are characterized by single neuron quasi-periodic dynamics and macroscopic periodic oscillation. In particular, in the studied systems each neuron is described by a simple "Leaky Integrate-and-Fire'' (LIF) model subjected to a forcing field, which represents the linear superposition of the pulses emitted in the past within the network.
These periodic oscillations of the field --- characteristic of the PS states --- resemble the Local Field Potential dynamics observed in real neural circuits --- like those measured in the cerebral cortex during epileptic seizures or the coherent oscillations recently recorded while monitoring the electric activity of cortical slices in the early days of postnatal rats. It must be stressed that in the latter case it is well know that the network is characterized by the presence of only excitatory synapses since inhibitory interactions are activated in the later stages of development of the rats.
A detailed investigation of the robustness and stability of the collective states with respect to the level of dilution of connectivity is thus fundamental since the cerebral cortex is characterized by neurons which are connected only to a fraction of the total number of cells. Preliminary results concerning LIF neural networks with impulsive excitatory coupling show that PS collective states are robust with respect to the dilution of connectivity, in particular, a random pruning in the number ofconnections of each neuron preserves these solutions which result in stable attractors of the system.
R. Zillmer, R. Livi, A. Politi, and A. Torcini "Desynchronization in diluted neural networks", Phys. Rev. E 74 (2006) 036203.
R. Zillmer, R. Livi, A. Politi, and A. Torcini "Desynchronized stable states in diluted neural networks", Neurocomputing 70 (2007) 1960.
R. Zillmer, R. Livi, A. Politi, and A. Torcini "Stability of the splay state in pulse--coupled networks" Phys. Rev. E 76 (2007) 046102
M. Calamai, A. Politi, and A. Torcini, ``Stability of splay states in globally coupled rotators'', Phys. Rev. E 80 (2009) 036209
A. Politi, and A. Torcini, ``Stable Chaos", Nonlinear Dynamics and Chaos: Advances and Perspectives Eds. Thiel, M.; Kurths, J.; Romano, M.C.: Moura, A.; Karolyi, G., (Springer Verlag, 2010, Heidelberg)
S. Olmi, R. Livi, A. Politi, and A. Torcini "Collective oscillations in disordered neural network", Phys. Rev. E 81 (2010) 046119
S. Luccioli and A. Politi, "Irregular Collective Behavior of Heterogeneous Neural Networks", Phys. Rev. Lett. 105, 158104 (2010)
A. Politi, S. Luccioli Dynamics of networks of leaky-integrate and fire neurons in "Complex Networks across the Natural and Technological Sciences" edited by M. Fox, D. Higham, G.-L. Oppo and E. Estrada, (Springer to appear).
S. Olmi, A. Politi, and A. Torcini, "Collective chaos in pulse coupled oscillators", submitted to EuroPhysLett in (2010) [PDF]