arXiv subjects

Zachary P. Kilpatrick

Publications and source records attributed to Zachary P. Kilpatrick.

25 records · Page 2Linked to original sources

arXiv2015

Stochastic synchronization of neural activity waves

We demonstrate that waves in distinct layers of a neuronal network can become phase-locked by common spatiotemporal noise. This phenomenon is studied for stationary bumps, traveling waves, and breathers. A weak noise expansion is used to derive an effective equation for the position of the wave in each layer, yielding a stochastic differential equation with multiplicative noise. Stability of the synchronous state is characterized by a Lyapunov exponent, which we can compute analytically from the reduced system. Our results extend previous work on limit-cycle oscillators, showing common noise can synchronize waves in a broad class of models.

nlin.PS↗

arXiv2015

Nonlinear Langevin equations for wandering patterns in stochastic neural fields

We analyze the effects of additive, spatially extended noise on spatiotemporal patterns in continuum neural fields. Our main focus is how fluctuations impact patterns when they are weakly coupled to an external stimulus or another equivalent pattern. Showing the generality of our approach, we study both propagating fronts and stationary bumps. Using a separation of time scales, we represent the effects of noise in terms of a phase-shift of a pattern from its uniformly translating position at long time scales, and fluctuations in the pattern profile around its instantaneous position at short time scales. In the case of a stimulus-locked front, we show that the phase-shift satisfies a nonlinear Langevin equation (SDE) whose deterministic part has a unique stable fixed point. Using a linear-noise approximation, we thus establish that wandering of the front about the stimulus-locked state is given by an Ornstein-Uhlenbeck (OU) process. Analogous results hold for the relative phase-shift between a pair of mutually coupled fronts, provided that the coupling is excitatory. On the other hand, if the mutual coupling is given by a Mexican hat function (difference of exponentials), then the linear-noise approximation breaks down due to the co-existence of stable and unstable phase-locked states in the deterministic limit. Similarly, the stochastic motion of mutually coupled bumps can be described by a system of nonlinearly coupled SDEs, which can be linearized to yield a multivariate OU process. As in the case of fronts, large deviations can cause bumps to temporarily decouple, leading to a phase-slip in the bump positions.

q-bio.NC↗

arXiv2014

Delay stabilizes stochastic motion of bumps in layered neural fields

We study the effects of propagation delays on the stochastic dynamics of bumps in neural fields with multiple layers. In the absence of noise, each layer supports a stationary bump. Using linear stability analysis, we show that delayed coupling between layers causes translating perturbations of the bumps to decay in the noise-free system. Adding noise to the system causes bumps to wander as a random walk. However, coupling between layers can reduce the variability of this stochastic motion by canceling noise that perturbs bumps in opposite directions. Delays in interlaminar coupling can further reduce variability, since they couple bump positions to states from the past. We demonstrate these relationships by deriving an asymptotic approximation for the effective motion of bumps. This yields a stochastic delay-differential equation where each delayed term arises from an interlaminar coupling. The impact of delays is well approximated by using a small delay expansion, which allows us to compute the effective diffusion in bumps' positions, accurately matching results from numerical simulations.

q-bio.NC↗

arXiv2014

Molecular mechanisms that regulate the coupled period of the mammalian circadian clock

In mammals, most cells in the brain and peripheral tissues generate circadian (~24hr) rhythms autonomously. These self-sustained rhythms are coordinated and entrained by a master circadian clock in the suprachiasmatic nucleus (SCN). Within the SCN, the individual rhythms of each neuron are synchronized through intercellular signaling. One important feature of SCN is that the synchronized period is close to the cell population mean of intrinsic periods. In this way, the synchronized period of the SCN stays close to the periods of cells in peripheral tissues. This is important for SCN to entrain cells throughout the body. However, the mechanism that drives the period of the coupled SCN cells to the population mean is not known. We use mathematical modeling and analysis to show that the mechanism of transcription repression plays a pivotal role in regulating the coupled period. Specifically, we use phase response curve analysis to show that the coupled period within the SCN stays near the population mean if transcriptional repression occurs via protein sequestration. In contrast, the coupled period is far from the mean if repression occurs through highly nonlinear Hill-type regulation (e.g. oligomer- or phosphorylation-based repression). Furthermore, we find that the timescale of intercellular coupling needs to be fast compared to that of intracellular feedback to maintain the mean period. These findings reveal the important relationship between the intracellular transcriptional feedback loop and intercellular coupling. This relationship explains why transcriptional repression appears to occur via protein sequestration in multicellular organisms, mammals and Drosophila, in contrast with the phosphorylation-based repression in unicellular organisms. That is, transition to protein sequestration is essential for synchronizing multiple cells with a period close to the population mean (~24hr).

q-bio.MN↗

arXiv2013

Coupling layers regularizes wave propagation in laminar stochastic neural fields

We study the effects of coupling between layers of stochastic neural field models with laminar structure. In particular, we focus on how the propagation of waves of neural activity in each layer is affected by the coupling. Synaptic connectivities within and between each layer are determined by integral kernels of an integrodifferential equation describing the temporal evolution of neural activity. Excitatory neural fields, with purely positive connectivities, support traveling fronts in each layer, whose speeds are increased when coupling between layers is considered. Studying the effects of noise, we find coupling also serves to reduce the variance in the position of traveling fronts, as long as the noise sources to each layer are not completely correlated. Neural fields with asymmetric connectivity support traveling pulses whose speeds are decreased by interlaminar coupling. Again, coupling reduces the variance in traveling pulse position, when noise is considered that is not totally correlated between layers. To derive our stochastic results, we employ a small-noise expansion, also assuming interlaminar connectivity scales similarly. Our asymptotic results agree reasonably with accompanying numerical simulations.

nlin.PS↗

arXiv2013

Optimizing working memory with heterogeneity of recurrent cortical excitation

A neural correlate of parametric working memory is a stimulus specific rise in neuron firing rate that persists long after the stimulus is removed. Network models with local excitation and broad inhibition support persistent neural activity, linking network architecture and parametric working memory. Cortical neurons receive noisy input fluctuations which causes persistent activity to diffusively wander about the network, degrading memory over time. We explore how cortical architecture that supports parametric working memory affects the diffusion of persistent neural activity. Studying both a spiking network and a simplified potential well model, we show that spatially heterogeneous excitatory coupling stabilizes a discrete number of persistent states, reducing the diffusion of persistent activity over the network. However, heterogeneous coupling also coarse-grains the stimulus representation space, limiting the capacity of parametric working memory. The storage errors due to coarse-graining and diffusion tradeoff so that information transfer between the initial and recalled stimulus is optimized at a fixed network heterogeneity. For sufficiently long delay times, the optimal number of attractors is less than the number of possible stimuli, suggesting that memory networks can under-represent stimulus space to optimize performance. Our results clearly demonstrate the effects of network architecture and stochastic fluctuations on parametric memory storage.

q-bio.NC↗

arXiv2012

Wandering bumps in stochastic neural fields

We study the effects of noise on stationary pulse solutions (bumps) in spatially extended neural fields. The dynamics of a neural field is described by an integrodifferential equation whose integral term characterizes synaptic interactions between neurons in different spatial locations of the network. Translationally symmetric neural fields support a continuum of stationary bump solutions, which may be centered at any spatial location. Random fluctuations are introduced by modeling the system as a spatially extended Langevin equation whose noise term we take to be multiplicative or additive. For nonzero noise, these bumps are shown to wander about the domain in a purely diffusive way. We can approximate the effective diffusion coefficient using a small noise expansion. Upon breaking the (continuous) translation symmetry of the system using a spatially heterogeneous inputs or synapses, bumps in the stochastic neural field can become temporarily pinned to a finite number of locations in the network. In the case of spatially heterogeneous synaptic weights, as the modulation frequency of this heterogeneity increases, the effective diffusion of bumps in the network approaches that of the network with spatially homogeneous weights.

nlin.PS↗