Low-dimensional firing-rate dynamics for populations of renewal-type spiking neurons

The macroscopic dynamics of large populations of neurons can be mathematically analyzed using low-dimensional firing-rate or neural-mass models. However, these models fail to capture spike synchronization effects of stochastic spiking neurons such as the non-stationary population response to rapidly changing stimuli. Here, we derive low-dimensional firing-rate models for homogeneous populations of general renewal-type neurons, including integrate-and-fire models driven by white noise. Renewal models account for neuronal refractoriness and spike synchronization dynamics. The derivation is based on an eigenmode expansion of the associated refractory density equation, which generalizes previous spectral methods for Fokker-Planck equations to arbitrary renewal models. We find a simple relation between the eigenvalues, which determine the characteristic time scales of the firing rate dynamics, and the Laplace transform of the interspike interval density or the survival function of the renewal process. Analytical expressions for the Laplace transforms are readily available for many renewal models including the leaky integrate-and-fire model. Retaining only the first eigenmode yields already an adequate low-dimensional approximation of the firing-rate dynamics that captures spike synchronization effects and fast transient dynamics at stimulus onset. We explicitly demonstrate the validity of our model for a large homogeneous population of Poisson neurons with absolute refractoriness, and other renewal models that admit an explicit analytical calculation of the eigenvalues. The here presented eigenmode expansion provides a systematic framework for novel firing-rate models in computational neuroscience based on spiking neuron dynamics with refractoriness.Comment: 24 pages, 7 figure

Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension

We investigate the motion of a run-and-tumble particle (RTP) in one dimension. We find the exact probability distribution of the particle with and without diffusion on the infinite line, as well as in a finite interval. In the infinite domain, this probability distribution approaches a Gaussian form in the long-time limit, as in the case of a regular Brownian particle. At intermediate times, this distribution exhibits unexpected multi-modal forms. In a finite domain, the probability distribution reaches a steady state form with peaks at the boundaries, in contrast to a Brownian particle. We also study the relaxation to the steady state analytically. Finally we compute the survival probability of the RTP in a semi-infinite domain. In the finite interval, we compute the exit probability and the associated exit times. We provide numerical verifications of our analytical results

Anomalous relaxation in colloidal systems

The Mpemba effect refers to a phenomenon where a sample of hot water may cool and begin to freeze more quickly than a cool or warm water sample prepared under identical conditions. Although the effect has been known since the time of Aristotle, it is named after the Tanzanian teenager Erasto Mpemba, who discovered the effect in the 1960s. Although Mpemba and Osborne showed the effect in laboratory experiments, it has always been mysterious, its underlying mechanism a topic of hot debate. In this thesis, we experimentally show the Mpemba effect in a colloidal system with a micron-sized silica bead diffusing in a bath. The bead is subjected to an external double-well potential created by a feedback-based optical tweezer. When a system is quenched from an initially hot equilibrium state to a cold equilibrium state, the evolution of the system between the initial and the final state is a strongly nonequilibrium process. As a nonequilibrium state cannot, in general, be characterized by a single temperature, we adopt the notion of a “distance” measure as a proxy for temperature. We show Mpemba effects in an asymmetric double-well potential. Our experimental results agree quantitatively with predictions based on the Fokker-Planck equation. Using understanding gained from the Mpemba effect, we design an experiment to investigate the opposite effect and present the first experimental evidence for this inverse Mpemba effect. Contrary to the cooling effect, the inverse effect is related to a phenomenon where a system that is initially cold heats up faster than an initially warm system. By understanding the underlying mechanism of these anomalous effects, we demonstrate strong Mpemba and inverse Mpemba effects, where a system can cool or heat exponentially faster to the bath temperature than under typical conditions. Finally, we ask whether asymmetry in the potential is necessary and show experimentally that an anomalous cooling effect can be observed in a symmetric potential, leading to a higher-order Mpemba effect

Quantum resetting in continuous measurement induced dynamics of a qubit

We study the evolution of a two-state system that is monitored continuously but with interactions with the detector tuned so as to avoid the Zeno affect. The system is allowed to interact with a sequence of prepared probes. The post-interaction probe states are measured and this leads to a stochastic evolution of the system's state vector, which can be described by a single angle variable. The system's effective evolution consists of a deterministic drift and a stochastic resetting to a fixed state at a rate that depends on the instantaneous state vector. The detector readout is a counting process. We obtain analytic results for the distribution of number of detector events and the time-evolution of the probability distribution. Earlier work on this model found transitions in the form of the steady state on increasing the measurement rate. Here we study transitions seen in the dynamics. As a spin-off we obtain, for a general stochastic resetting process with diffusion, drift and position dependent jump rates, an exact and general solution for the evolution of the probability distribution.Comment: 27 pages, 4 figure

Oscillatory Dynamics in Rock-Paper-Scissors Games with Mutations

We study the oscillatory dynamics in the generic three-species rock-paper-scissors games with mutations. In the mean-field limit, different behaviors are found: (a) for high mutation rate, there is a stable interior fixed point with coexistence of all species; (b) for low mutation rates, there is a region of the parameter space characterized by a limit cycle resulting from a Hopf bifurcation; (c) in the absence of mutations, there is a region where heteroclinic cycles yield oscillations of large amplitude (not robust against noise). After a discussion on the main properties of the mean-field dynamics, we investigate the stochastic version of the model within an individual-based formulation. Demographic fluctuations are therefore naturally accounted and their effects are studied using a diffusion theory complemented by numerical simulations. It is thus shown that persistent erratic oscillations (quasi-cycles) of large amplitude emerge from a noise-induced resonance phenomenon. We also analytically and numerically compute the average escape time necessary to reach a (quasi-)cycle on which the system oscillates at a given amplitude.Comment: 25 pages, 9 figures. To appear in the Journal of Theoretical Biolog

Master equations

The dynamics of a complex physical, biological, or chemical systems can often be modelled in terms of a continuous-time Markov process. The governing equations of these processes are the Fokker-Planck and the master equation. Both equations assume that the future of a system depends only on its current state, memories of its past having been wiped out by randomizing forces. Whereas the Fokker-Planck equation describes a system that evolves continuously from one state to another, the master equation models a system that performs jumps between its states. In this thesis, we focus on master equations. We first present a comprehensive mathematical framework for the analytical and numerical analysis of master equations in chapter I. Special attention is given to their representation by path integrals. In the subsequent chapters, master equations are applied to the study of physical and biological systems. In chapter II, we study the stochastic and deterministic evolution of zero-sum games and thereby explain a condensation phenomenon expected in driven-dissipative bosonic quantum systems. Afterwards, in chapter III, we develop a coarse-grained model of microbial range expansions and use it to predict which of three strains of Escherichia coli survive such an expansion