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.

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.

For a class of integrate-and-fire, pulse-coupled networks with complex topology, we study the dependence of the pulse rate on the underlying architectural connectivity statistics. We derive the distribution of the pulse rate from this dependence and determine when the underlying scale-free architectural connectivity gives rise to a scale-free pulse-rate distribution. We identify the scaling of the pairwise coupling between the dynamical units in this network class that keeps their pulse rates bounded in the infinite-network limit. In the process, we determine the connectivity statistics for a specific scale-free network grown by preferential attachment.

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. © 2007 Optical Society of America OCIS codes: 060.2310, 190.4370, 060.2330 Bisolitons in optical fiber lines with dispersion management were first discovered by using computer modeling 1 and later were discovered experimentally. 2The former were in-phase and the latter were antiphase bisolitons, which can be viewed as twocomponent soliton molecules. In numerical simulations, they are stable over long propagation distances and, if perturbed, oscillate about equilibrium. Following the experimental work we investigate the structure of antiphase bisolitons, assuming that fiber losses are completely compensated and propagation of pulses through optical fiber in a dispersionmanaged system is governed by the nonlinear Schrödinger equationwhere u = u͑t , z͒ is the slowly varying envelope of the electromagnetic field inside the fiber. We consider a simple case of a piecewise constant dispersion function d͑z͒, where a fiber span of length z dm / 2 with normal dispersion alternates with equal-length spans of anomalous dispersion fiber. The function d͑z͒ can be represented as a sum of an oscillating part d ͑z͒ and aIn this system, the characteristic length of the nonlinearity is z nl ϳ 1/␥P, where P is the peak power of the bisoliton, while the characteristic length of the residual dispersion is z d 0 ϳ 2 / d 0 , where is the pulse width. The spectrum û of the solution to Eq. (1) can be represented asprovided that z dm z nl , z d 0 . The exponential term captures the fast (in z) phase, and q͑ , z͒ captures the slow amplitude dynamics of the spectral components. As has been shown, 3 the evolution of the spectral components at leading order can be described bywhereHere s ϵ z dm d 1 / 2 is dispersion map strength and ⌬ ϵ 1 2 + 2 2 − 3 2 − 2 . Higher-order corrections to this equation were considered in Ref. 4 to take into account the effect of dispersion map geometry on inphase bisolitons. We determine a shape of a bisoliton solution following earlier work by Lushnikov.5 If a solitary wave solution with phase period −1 has the form q͑͒ = A͑͒e iz , then amplitude A͑͒ evolves according to the following integral equation:Rescaling variables t = 0 , = ⍀ / 0 , and A͑͒ = a͑⍀͒, where 0 = s 1/2 and a =2͑s / ␥͒ 1/2 , results in a dimensionless equationwherewhich depends on a single parameter d 0 = d 0 / ͑s͒. Here ⌬ = ⍀ 1 2 + ⍀ 2 2 − ⍀ 3 2 − ⍀ 2 . We study the structure of antiphase bisolitons as a function of d 0 . To solve this integral equation we use the following iterative procedure: March 15, 2007 / Vol. 32, No. 6 / OPTICS LETTERS 6050146-9592/07/060605-3/$15.00

Using integrate-and-fire networks, we study the relationship between the architectural connectivity of a network and its functional connectivity as characterized by the network's 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.