Jay M. Newby

Metastability in a Stochastic Neural Network Modeled as a Velocity Jump Markov Process

One of the major challenges in neuroscience is to determine how noise that is present at the molecular and cellular levels affects dynamics and information processing at the macroscopic level of synapti- cally coupled neuronal populations. Often noise is incorporated into deterministic network models using extrinsic noise sources. An alternative approach is to assume that noise arises intrinsically as a collective population effect, which has led to a master equation formulation of stochastic neural networks. In this paper we extend the master equation formulation by introducing a stochastic model of neural population dynamics in the form of a velocity jump Markov process. The latter has the advantage of keeping track of synaptic processing as well as spiking activity, and reduces to the neural master equation in a particular limit. The population synaptic variables evolve ac- cording to piecewise deterministic dynamics, which depends on population spiking activity. The latter is characterized by a set of discrete stochastic variables evolving according to a jump Markov process, with transition rates that depend on the synaptic variables. We consider the particular problem of rare transitions between metastable states of a network operating in a bistable regime in the deterministic limit. Assuming that the synaptic dynamics is much slower than the transitions between discrete spiking states, we use a WKB approximation and singular perturbation theory to determine the mean first passage time to cross the separatrix between the two metastable states. Such an analysis can also be applied to other velocity jump Markov processes, including stochastic voltage-gated ion channels and stochastic gene networks.

The Binding Site Barrier Elicited by Tumor-Associated Fibroblasts Interferes Disposition of Nanoparticles in Stroma-Vessel Type Tumors

The binding site barrier (BSB) was originally proposed to describe the binding behavior of antibodies to cells peripheral to blood vessels, preventing their further penetration into the tumors. Yet, it is revisited herein to describe the intratumoral cellular disposition of nanoparticles (NPs). Specifically, the BSB limits NP diffusion and results in unintended internalization of NPs by stroma cells localized near blood vessels. This not only limits the therapeutic outcome but also promotes adverse off-target effects. In the current study, it was shown that tumor-associated fibroblast cells (TAFs) are the major component of the BSB, particularly in tumors with a stroma-vessel architecture where the location of TAFs aligns with blood vessels. Specifically, TAF distance to blood vessels, expression of receptor proteins, and binding affinity affect the intensity of the BSB. The physical barrier elicited by extracellular matrix also prolongs the retention of NPs in the stroma, potentially contributing to the BSB. The influence of particle size on the BSB was also investigated. The strongest BSB effect was found with small (∼18 nm) NPs targeted with the anisamide ligand. The uptake of these NPs by TAFs was about 7-fold higher than that of the other cells 16 h post-intravenous injection. This was because TAFs also expressed the sigma receptor under the influence of TGF-β secreted by the tumor cells. Overall, the current study underscores the importance of BSBs in the delivery of nanotherapeutics and provides a rationale for exploiting BSBs to target TAFs.

Isolating intrinsic noise sources in a stochastic genetic switch.

The stochastic mutual repressor model is analysed using perturbation methods. This simple model of a gene circuit consists of two genes and three promotor states. Either of the two protein products can dimerize, forming a repressor molecule that binds to the promotor of the other gene. When the repressor is bound to a promotor, the corresponding gene is not transcribed and no protein is produced. Either one of the promotors can be repressed at any given time or both can be unrepressed, leaving three possible promotor states. This model is analysed in its bistable regime in which the deterministic limit exhibits two stable fixed points and an unstable saddle, and the case of small noise is considered. On small timescales, the stochastic process fluctuates near one of the stable fixed points, and on large timescales, a metastable transition can occur, where fluctuations drive the system past the unstable saddle to the other stable fixed point. To explore how different intrinsic noise sources affect these transitions, fluctuations in protein production and degradation are eliminated, leaving fluctuations in the promotor state as the only source of noise in the system. The process without protein noise is then compared to the process with weak protein noise using perturbation methods and Monte Carlo simulations. It is found that some significant differences in the random process emerge when the intrinsic noise source is removed.

Quasi-steady State Reduction of Molecular Motor-Based Models of Directed Intermittent Search

We present a quasi-steady state reduction of a linear reaction-hyperbolic master equation describing the directed intermittent search for a hidden target by a motor-driven particle moving on a one-dimensional filament track. The particle is injected at one end of the track and randomly switches between stationary search phases and mobile nonsearch phases that are biased in the anterograde direction. There is a finite possibility that the particle fails to find the target due to an absorbing boundary at the other end of the track. Such a scenario is exemplified by the motor-driven transport of vesicular cargo to synaptic targets located on the axon or dendrites of a neuron. The reduced model is described by a scalar Fokker–Planck (FP) equation, which has an additional inhomogeneous decay term that takes into account absorption by the target. The FP equation is used to compute the probability of finding the hidden target (hitting probability) and the corresponding conditional mean first passage time (MFPT) in terms of the effective drift velocity V, diffusivity D, and target absorption rate λ of the random search. The quasi-steady state reduction determines V, D, and λ in terms of the various biophysical parameters of the underlying motor transport model. We first apply our analysis to a simple 3-state model and show that our quasi-steady state reduction yields results that are in excellent agreement with Monte Carlo simulations of the full system under physiologically reasonable conditions. We then consider a more complex multiple motor model of bidirectional transport, in which opposing motors compete in a “tug-of-war”, and use this to explore how ATP concentration might regulate the delivery of cargo to synaptic targets.

Breakdown of Fast-Slow Analysis in an Excitable System with Channel Noise

We consider a stochastic version of an excitable system based on the Morris-Lecar model of a neuron, in which the noise originates from stochastic sodium and potassium ion channels opening and closing. One can analyze neural excitability in the deterministic model by using a separation of time scales involving a fast voltage variable and a slow recovery variable, which represents the fraction of open potassium channels. In the stochastic setting, spontaneous excitation is initiated by ion channel noise. If the recovery variable is constant during initiation, the spontaneous activity rate can be calculated using Kramers rate theory. The validity of this assumption in the stochastic model is examined using a systematic perturbation analysis. We find that, in most physically relevant cases, this assumption breaks down, requiring an alternative to Kramers theory for excitable systems with one deterministic fixed point. We also show that an exit time problem can be formulated in an excitable system by considering maximum likelihood trajectories of the stochastic process.

Effects of moderate noise on a limit cycle oscillator: counterrotation and bistability.

The effects of noise on the dynamics of nonlinear systems is known to lead to many counterintuitive behaviors. Using simple planar limit cycle oscillators, we show that the addition of moderate noise leads to qualitatively different dynamics. In particular, the system can appear bistable, rotate in the opposite direction of the deterministic limit cycle, or cease oscillating altogether. Utilizing standard techniques from stochastic calculus and recently developed stochastic phase reduction methods, we elucidate the mechanisms underlying the different dynamics and verify our analysis with the use of numerical simulations. Last, we show that similar bistable behavior is found when moderate noise is applied to the FitzHugh-Nagumo model, which is more commonly used in biological applications.

Local synaptic signaling enhances the stochastic transport of motor-driven cargo in neurons

The tug-of-war model of motor-driven cargo transport is formulated as an intermittent trapping process. An immobile trap, representing the cellular machinery that sequesters a motor-driven cargo for eventual use, is located somewhere within a microtubule track. A particle representing a motor-driven cargo that moves randomly with a forward bias is introduced at the beginning of the track. The particle switches randomly between a fast moving phase and a slow moving phase. When in the slow moving phase, the particle can be captured by the trap. To account for the possibility that the particle avoids the trap, an absorbing boundary is placed at the end of the track. Two local signaling mechanisms--intended to improve the chances of capturing the target--are considered by allowing the trap to affect the tug-of-war parameters within a small region around itself. The first is based on a localized adenosine triphosphate (ATP) concentration gradient surrounding a synapse, and the second is based on a concentration of tau--a microtubule-associated protein involved in Alzheimers disease--coating the microtubule near the synapse. It is shown that both mechanisms can lead to dramatic improvements in the capture probability, with a minimal increase in the mean capture time. The analysis also shows that tau can cause a cargo to undergo random oscillations, which could explain some experimental observations.

An Asymptotic Analysis of the Spatially Inhomogeneous Velocity-Jump Process

We analyze the one-dimensional velocity-jump process, where a particle moves at a constant velocity determined by the particle’s internal velocity state that randomly fluctuates with exponentially distributed waiting times. The transition rates between the internal velocity states depend on the location of the particle, leading to a spatially inhomogeneous random process. An asymptotic analysis is applied to obtain the stationary distribution of the random process. The result is compared to the often-used quasi–steady-state diffusion approximation, and it is found that the diffusion approximation breaks down in the presence of a turning point, where the average velocity of the particle changes sign. We extend the analysis to approximate the first-exit time density for the particle to escape the confining effect of the turning point, and we find the diffusion approximation also fails to accurately describe the long-time behavior of the process. The accuracy of the two approximations is explored for a simpl...

Directed intermittent search for hidden targets

We develop and analyze a stochastic model of directed intermittent search for a hidden target on a one-dimensional track. A particle injected at one end of the track randomly switches between a stationary search phase and a mobile, non-search phase that is biased in the anterograde direction. There is a finite possibility that the particle fails to find the target due to an absorbing boundary at the other end of the track or due to competition with other targets. Such a scenario is exemplified by the motor-driven transport of mRNA granules to synaptic targets along a dendrite. We first calculate the hitting probability and conditional mean first passage time (MFPT) for finding a single target. We show that an optimal search strategy does not exist, although for a fixed hitting probability, a unidirectional rather than a partially biased search strategy generates a smaller MFPT. We then extend our analysis to the case of multiple targets, and determine how the hitting probability and MFPT depend on the number of targets.

Bistable switching asymptotics for the self regulating gene

A simple stochastic model of a self regulating gene that displays bistable switching is analyzed. While on, a gene transcribes mRNA at a constant rate. Transcription factors can bind to the DNA and affect the genes transcription rate. Before an mRNA is degraded, it synthesizes protein, which in turn regulates gene activity by influencing the activity of transcription factors. Protein is slowly removed from the system through degradation. Depending on how the protein regulates gene activity, the protein concentration can exhibit noise induced bistable switching. An asymptotic approximation of the mean switching rate is derived that includes the pre exponential factor, which improves upon a previously reported logarithmically accurate approximation. With the improved accuracy, a uniformly accurate approximation of the stationary probability density, describing the gene, mRNA copy number, and protein concentration is also obtained.