Maxim S. Shkarayev

Twin families of bisolitons in dispersion-managed systems

We calculate bisoliton solutions by using a slowly varying stroboscopic equation. The system is characterized in terms of a single dimensionless parameter. We find two branches of solutions and describe the structure of the tails for the lower-branch solutions.

Epidemics in adaptive social networks with temporary link deactivation

Disease spread in a society depends on the topology of the network of social contacts. Moreover, individuals may respond to the epidemic by adapting their contacts to reduce the risk of infection, thus changing the network structure and affecting future disease spread. We propose an adaptation mechanism where healthy individuals may choose to temporarily deactivate their contacts with sick individuals, allowing reactivation once both individuals are healthy. We develop a mean-field description of this system and find two distinct regimes: slow network dynamics, where the adaptation mechanism simply reduces the effective number of contacts per individual, and fast network dynamics, where more efficient adaptation reduces the spread of disease by targeting dangerous connections. Analysis of the bifurcation structure is supported by numerical simulations of disease spread on an adaptive network. The system displays a single parameter-dependent stable steady state and non-monotonic dependence of connectivity on link deactivation rate.

Epidemics with temporary link deactivation in scale-free networks

During an epidemic, people may adapt or alter their social contacts to avoid infection. Various adaptation mechanisms have been studied previously. Recently, a new adaptation mechanism was presented in [1], where susceptible nodes temporarily deactivate their links to infected neighbors and reactivate when their neighbors recover. Considering the same adaptation mechanism on a scale-free network, we find that the topology of the subnetwork consisting of active links is fundamentally different from the original network topology. We predict the scaling exponent of the active degree distribution and derive mean-field equations by using improved moment closure approximations based on the conditional distribution of active degree given the total degree. These mean field equations show better agreement with numerical simulation results than the standard mean field equations based on a homogeneity assumption.

Dynamics of the exponential integrate-and-fire model with slow currents and adaptation

In order to properly capture spike-frequency adaptation with a simplified point-neuron model, we study approximations of Hodgkin-Huxley (HH) models including slow currents by exponential integrate-and-fire (EIF) models that incorporate the same types of currents. We optimize the parameters of the EIF models under the external drive consisting of AMPA-type conductance pulses using the current-voltage curves and the van Rossum metric to best capture the subthreshold membrane potential, firing rate, and jump size of the slow current at the neuron’s spike times. Our numerical simulations demonstrate that, in addition to these quantities, the approximate EIF-type models faithfully reproduce bifurcation properties of the HH neurons with slow currents, which include spike-frequency adaptation, phase-response curves, critical exponents at the transition between a finite and infinite number of spikes with increasing constant external drive, and bifurcation diagrams of interspike intervals in time-periodically forced models. Dynamics of networks of HH neurons with slow currents can also be approximated by corresponding EIF-type networks, with the approximation being at least statistically accurate over a broad range of Poisson rates of the external drive. For the form of external drive resembling realistic, AMPA-like synaptic conductance response to incoming action potentials, the EIF model affords great savings of computation time as compared with the corresponding HH-type model. Our work shows that the EIF model with additional slow currents is well suited for use in large-scale, point-neuron models in which spike-frequency adaptation is important.

Exact results for a simple epidemic model on a directed network: Explorations of a system in a nonequilibrium steady state

Motivated by fundamental issues in nonequilibrium statistical mechanics, we study the venerable susceptible-infected-susceptible (SIS) model of disease spreading in an idealized, simple setting. Using Monte Carlo and analytic techniques, we consider a fully connected, unidirectional network of odd number of nodes, each having an equal number of in- and out-degrees. With the standard SIS dynamics at high infection rates, this system settles into an active nonequilibrium steady state. We find the exact probability distribution and explore its implications for nonequilibrium statistical mechanics, such as the presence of persistent probability currents.

Recruitment dynamics in adaptive social networks

We model recruitment in adaptive social networks in the presence of birth and death processes. Recruitment is characterized by nodes changing their status to that of the recruiting class as a result of contact with recruiting nodes. Only a susceptible subset of nodes can be recruited. The recruiting individuals may adapt their connections in order to improve recruitment capabilities, thus changing the network structure adaptively. We derive a mean field theory to predict the dependence of the growth threshold of the recruiting class on the adaptation parameter. Furthermore, we investigate the effect of adaptation on the recruitment level, as well as on network topology. The theoretical predictions are compared with direct simulations of the full system. We identify two parameter regimes with qualitatively different bifurcation diagrams depending on whether nodes become susceptible frequently (multiple times in their lifetime) or rarely (much less than once per lifetime).

Architectural and functional connectivity in scale-free integrate-and-fire networks

Using integrate-and-fire networks, we study the relationship between the architectural connectivity of a network and its functional connectivity as characterized by the networks dynamical properties. We show that dynamics on a complex network can be controlled by the topology of the network, in particular, scale-free functional connectivity can arise from scale-free architectural connectivity, in which the architectural degree correlation plays a crucial role.

Asymptotically inspired moment-closure approximation for adaptive networks

Adaptive social networks, in which nodes and network structure coevolve, are often described using a mean-field system of equations for the density of node and link types. These equations constitute an open system due to dependence on higher-order topological structures. We propose a new approach to moment closure based on the analytical description of the system in an asymptotic regime. We apply the proposed approach to two examples of adaptive networks: recruitment to a cause model and adaptive epidemic model. We show a good agreement between the improved mean-field prediction and simulations of the full network system.