134,322 research outputs found
Dynamical Systems on Networks: A Tutorial
We give a tutorial for the study of dynamical systems on networks. We focus
especially on "simple" situations that are tractable analytically, because they
can be very insightful and provide useful springboards for the study of more
complicated scenarios. We briefly motivate why examining dynamical systems on
networks is interesting and important, and we then give several fascinating
examples and discuss some theoretical results. We also briefly discuss
dynamical systems on dynamical (i.e., time-dependent) networks, overview
software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than
original version, some reorganization and also more pointers to interesting
direction
Intrinsically-generated fluctuating activity in excitatory-inhibitory networks
Recurrent networks of non-linear units display a variety of dynamical regimes
depending on the structure of their synaptic connectivity. A particularly
remarkable phenomenon is the appearance of strongly fluctuating, chaotic
activity in networks of deterministic, but randomly connected rate units. How
this type of intrinsi- cally generated fluctuations appears in more realistic
networks of spiking neurons has been a long standing question. To ease the
comparison between rate and spiking networks, recent works investigated the
dynami- cal regimes of randomly-connected rate networks with segregated
excitatory and inhibitory populations, and firing rates constrained to be
positive. These works derived general dynamical mean field (DMF) equations
describing the fluctuating dynamics, but solved these equations only in the
case of purely inhibitory networks. Using a simplified excitatory-inhibitory
architecture in which DMF equations are more easily tractable, here we show
that the presence of excitation qualitatively modifies the fluctuating activity
compared to purely inhibitory networks. In presence of excitation,
intrinsically generated fluctuations induce a strong increase in mean firing
rates, a phenomenon that is much weaker in purely inhibitory networks.
Excitation moreover induces two different fluctuating regimes: for moderate
overall coupling, recurrent inhibition is sufficient to stabilize fluctuations,
for strong coupling, firing rates are stabilized solely by the upper bound
imposed on activity, even if inhibition is stronger than excitation. These
results extend to more general network architectures, and to rate networks
receiving noisy inputs mimicking spiking activity. Finally, we show that
signatures of the second dynamical regime appear in networks of
integrate-and-fire neurons
Transition to chaos in random neuronal networks
Firing patterns in the central nervous system often exhibit strong temporal
irregularity and heterogeneity in their time averaged response properties.
Previous studies suggested that these properties are outcome of an intrinsic
chaotic dynamics. Indeed, simplified rate-based large neuronal networks with
random synaptic connections are known to exhibit sharp transition from fixed
point to chaotic dynamics when the synaptic gain is increased. However, the
existence of a similar transition in neuronal circuit models with more
realistic architectures and firing dynamics has not been established.
In this work we investigate rate based dynamics of neuronal circuits composed
of several subpopulations and random connectivity. Nonzero connections are
either positive-for excitatory neurons, or negative for inhibitory ones, while
single neuron output is strictly positive; in line with known constraints in
many biological systems. Using Dynamic Mean Field Theory, we find the phase
diagram depicting the regimes of stable fixed point, unstable dynamic and
chaotic rate fluctuations. We characterize the properties of systems near the
chaotic transition and show that dilute excitatory-inhibitory architectures
exhibit the same onset to chaos as a network with Gaussian connectivity.
Interestingly, the critical properties near transition depend on the shape of
the single- neuron input-output transfer function near firing threshold.
Finally, we investigate network models with spiking dynamics. When synaptic
time constants are slow relative to the mean inverse firing rates, the network
undergoes a sharp transition from fast spiking fluctuations and static firing
rates to a state with slow chaotic rate fluctuations. When the synaptic time
constants are finite, the transition becomes smooth and obeys scaling
properties, similar to crossover phenomena in statistical mechanicsComment: 28 Pages, 12 Figures, 5 Appendice
Collective stability of networks of winner-take-all circuits
The neocortex has a remarkably uniform neuronal organization, suggesting that
common principles of processing are employed throughout its extent. In
particular, the patterns of connectivity observed in the superficial layers of
the visual cortex are consistent with the recurrent excitation and inhibitory
feedback required for cooperative-competitive circuits such as the soft
winner-take-all (WTA). WTA circuits offer interesting computational properties
such as selective amplification, signal restoration, and decision making. But,
these properties depend on the signal gain derived from positive feedback, and
so there is a critical trade-off between providing feedback strong enough to
support the sophisticated computations, while maintaining overall circuit
stability. We consider the question of how to reason about stability in very
large distributed networks of such circuits. We approach this problem by
approximating the regular cortical architecture as many interconnected
cooperative-competitive modules. We demonstrate that by properly understanding
the behavior of this small computational module, one can reason over the
stability and convergence of very large networks composed of these modules. We
obtain parameter ranges in which the WTA circuit operates in a high-gain
regime, is stable, and can be aggregated arbitrarily to form large stable
networks. We use nonlinear Contraction Theory to establish conditions for
stability in the fully nonlinear case, and verify these solutions using
numerical simulations. The derived bounds allow modes of operation in which the
WTA network is multi-stable and exhibits state-dependent persistent activities.
Our approach is sufficiently general to reason systematically about the
stability of any network, biological or technological, composed of networks of
small modules that express competition through shared inhibition.Comment: 7 Figure
Flexible Memory Networks
Networks of neurons in some brain areas are flexible enough to encode new
memories quickly. Using a standard firing rate model of recurrent networks, we
develop a theory of flexible memory networks. Our main results characterize
networks having the maximal number of flexible memory patterns, given a
constraint graph on the network's connectivity matrix. Modulo a mild
topological condition, we find a close connection between maximally flexible
networks and rank 1 matrices. The topological condition is H_1(X;Z)=0, where X
is the clique complex associated to the network's constraint graph; this
condition is generically satisfied for large random networks that are not
overly sparse. In order to prove our main results, we develop some
matrix-theoretic tools and present them in a self-contained section independent
of the neuroscience context.Comment: Accepted to Bulletin of Mathematical Biology, 11 July 201
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