Circuits and Mechanisms for Surround Modulation in Visual Cortex.

Surround modulation (SM) is a fundamental property of sensory neurons in many species and sensory modalities. SM is the ability of stimuli in the surround of a neuron's receptive field (RF) to modulate (typically suppress) the neuron's response to stimuli simultaneously presented inside the RF, a property thought to underlie optimal coding of sensory information and important perceptual functions. Understanding the circuit and mechanisms for SM can reveal fundamental principles of computations in sensory cortices, from mouse to human. Current debate is centered over whether feedforward or intracortical circuits generate SM, and whether this results from increased inhibition or reduced excitation. Here we present a working hypothesis, based on theoretical and experimental evidence, that SM results from feedforward, horizontal, and feedback interactions with local recurrent connections, via synaptic mechanisms involving both increased inhibition and reduced recurrent excitation. In particular, strong and balanced recurrent excitatory and inhibitory circuits play a crucial role in the computation of SM.

Asymptotic analysis of particle cluster formation in the presence of anchoring sites.

The aggregation or clustering of proteins and other macromolecules plays an important role in the formation of large-scale molecular assemblies within cell membranes. Examples of such assemblies include lipid rafts, and postsynaptic domains (PSDs) at excitatory and inhibitory synapses in neurons. PSDs are rich in scaffolding proteins that can transiently trap transmembrane neurotransmitter receptors, thus localizing them at specific spatial positions. Hence, PSDs play a key role in determining the strength of synaptic connections and their regulation during learning and memory. Recently, a two-dimensional (2D) diffusion-mediated aggregation model of PSD formation has been developed in which the spatial locations of the clusters are determined by a set of fixed anchoring sites. The system is kept out of equilibrium by the recycling of particles between the cell membrane and interior. This results in a stationary distribution consisting of multiple clusters, whose average size can be determined using an effective mean-field description of the particle concentration around each anchored cluster. In this paper, we derive corrections to the mean-field approximation by applying the theory of diffusion in singularly perturbed domains. The latter is a powerful analytical method for solving two-dimensional (2D) and three-dimensional (3D) diffusion problems in domains where small holes or perforations have been removed from the interior. Applications range from modeling intracellular diffusion, where interior holes could represent subcellular structures such as organelles or biological condensates, to tracking the spread of chemical pollutants or heat from localized sources. In this paper, we take the bounded domain to be the cell membrane and the holes to represent anchored clusters. The analysis proceeds by partitioning the membrane into a set of inner regions around each cluster, and an outer region where mean-field interactions occur. Asymptotically matching the inner and outer stationary solutions generates an asymptotic expansion of the particle concentration, which includes higher-order corrections to mean-field theory that depend on the positions of the clusters and the boundary of the domain. Motivated by a recent study of light-activated protein oligomerization in cells, we also develop the analogous theory for cluster formation in a three-dimensional (3D) domain. The details of the asymptotic analysis differ from the 2D case due to the contrasting singularity structure of 2D and 3D Green's functions.

Global density equations for interacting particle systems with stochastic resetting: From overdamped Brownian motion to phase synchronization.

A wide range of phenomena in the natural and social sciences involve large systems of interacting particles, including plasmas, collections of galaxies, coupled oscillators, cell aggregations, and economic "agents." Kinetic methods for reducing the complexity of such systems typically involve the derivation of nonlinear partial differential equations for the corresponding global densities. In recent years, there has been considerable interest in the mean field limit of interacting particle systems with long-range interactions. Two major examples are interacting Brownian particles in the overdamped regime and the Kuramoto model of coupled phase oscillators. In this paper, we analyze these systems in the presence of local or global stochastic resetting, where the position or phase of each particle independently or simultaneously resets to its original value at a random sequence of times generated by a Poisson process. In each case, we derive the Dean-Kawasaki (DK) equation describing hydrodynamic fluctuations of the global density and then use a mean field ansatz to obtain the corresponding nonlinear McKean-Vlasov (MV) equation in the thermodynamic limit. In particular, we show how the MV equation for global resetting is driven by a Poisson noise process, reflecting the fact that resetting is common to all of the particles and, thus, induces correlations that cannot be eliminated by taking a mean field limit. We then investigate the effects of local and global resetting on nonequilibrium stationary solutions of the macroscopic dynamics and, in the case of the Kuramoto model, the reduced dynamics on the Ott-Antonsen manifold.

Morphogen gradient formation in partially absorbing media.

Morphogen gradients play an essential role in the spatial regulation of cell patterning during early development. The classical mechanism of morphogen gradient formation involves the diffusion of morphogens away from a localized source combined with some form of bulk absorption. Morphogen gradient formation plays a crucial role during early development, whereby a spatially varying concentration of morphogen protein drives a corresponding spatial variation in gene expression during embryogenesis. In most models, the absorption rate is taken to be a constant multiple of the local concentration. In this paper, we explore a more general class of diffusion-based model in which absorption is formulated probabilistically in terms of a stopping time condition. Absorption of each particle occurs when its time spent within the bulk domain (occupation time) exceeds a randomly distributed thresholda; the classical model with a constant rate of absorption is recovered by taking the threshold distributionΨ(a)=e-κ0a. We explore how the choice of Ψ(a) affects the steady-state concentration gradient, and the relaxation to steady-state as determined by the accumulation time. In particular, we show that the more general model can generate similar concentration profiles to the classical case, while significantly reducing the accumulation time.

Stochastic neural field model of stimulus-dependent variability in cortical neurons.

We use stochastic neural field theory to analyze the stimulus-dependent tuning of neural variability in ring attractor networks. We apply perturbation methods to show how the neural field equations can be reduced to a pair of stochastic nonlinear phase equations describing the stochastic wandering of spontaneously formed tuning curves or bump solutions. These equations are analyzed using a modified version of the bivariate von Mises distribution, which is well-known in the theory of circular statistics. We first consider a single ring network and derive a simple mathematical expression that accounts for the experimentally observed bimodal (or M-shaped) tuning of neural variability. We then explore the effects of inter-network coupling on stimulus-dependent variability in a pair of ring networks. These could represent populations of cells in two different layers of a cortical hypercolumn linked via vertical synaptic connections, or two different cortical hypercolumns linked by horizontal patchy connections within the same layer. We find that neural variability can be suppressed or facilitated, depending on whether the inter-network coupling is excitatory or inhibitory, and on the relative strengths and biases of the external stimuli to the two networks. These results are consistent with the general observation that increasing the mean firing rate via external stimuli or modulating drives tends to reduce neural variability.

Stochastic Turing Pattern Formation in a Model with Active and Passive Transport.

We investigate Turing pattern formation in a stochastic and spatially discretized version of a reaction-diffusion-advection (RDA) equation, which was previously introduced to model synaptogenesis in C. elegans. The model describes the interactions between a passively diffusing molecular species and an advecting species that switches between anterograde and retrograde motor-driven transport (bidirectional transport). Within the context of synaptogenesis, the diffusing molecules can be identified with the protein kinase CaMKII and the advecting molecules as glutamate receptors. The stochastic dynamics evolves according to an RDA master equation, in which advection and diffusion are both modeled as hopping reactions along a one-dimensional array of chemical compartments. Carrying out a linear noise approximation of the RDA master equation leads to an effective Langevin equation, whose power spectrum provides a means of extending the definition of a Turing instability to stochastic systems, namely in terms of the existence of a peak in the power spectrum at a nonzero spatial frequency. We thus show how noise can significantly extend the range over which spontaneous patterns occur, which is consistent with previous studies of RD systems.

Impulsive signaling model of cytoneme-based morphogen gradient formation.

Morphogen protein gradients play a vital role in regulating spatial pattern formation during development. The most commonly accepted mechanism of protein gradient formation involves the diffusion and degradation of morphogens from a localized source. However, there is growing experimental evidence for a direct cell-to-cell signaling mechanism via thin actin-rich cellular extensions known as cytonemes. Recent modeling studies of cytoneme-based morphogenesis in invertebrates ignore the discrete nature of vesicular transport along cytonemes, focusing on deterministic continuum models. In this paper, we develop an impulsive signaling model of morphogen gradient formation in invertebrates, which takes into account the discrete and stochastic nature of vesicular transport along cytonemes. We begin by solving a first passage time problem with sticky boundaries to determine the expected time to deliver a vesicle to a target cell, assuming that there is a 'nucleation' time for injecting the vesicle into the cytoneme. We then use queuing theory to analyze the impulsive model of morphogen gradient formation in the case of multiple cytonemes and multiple targets. In particular, we determine the steady-state mean and variance of the morphogen distribution across a one-dimensional array of target cells. The mean distribution recovers the spatially decaying morphogen gradient of previous deterministic models. However, the burst-like nature of morphogen transport can lead to Fano factors greater than unity across the array of cells, resulting in significant fluctuations at more distant target sites.

Modeling the Role of Feedback in the Adaptive Response of Bacterial Quorum Sensing.

Bacterial quorum sensing (QS) is a form of intercellular communication that relies on the production and detection of diffusive signaling molecules called autoinducers. Such a mechanism allows the bacteria to track their cell density in order to regulate group behavior, such as biofilm formation and bioluminescence. In a number of bacterial QS systems, including V. harveyi, multiple signaling pathways are integrated into a single phosphorylation-dephosphorylation cycle. In this paper, we propose a weight control mechanism, in which QS uses feedback loops to 'decode' the integrated signals by actively changing the sensitivity in different pathways. We first use a slow/fast analysis to reduce a single-cell model to a planar dynamical system involving the concentrations of phosphorylated signaling protein LuxU and a small non-coding RNA. In addition to identifying the weight control mechanism, we show that adding a feedback loop can lead to a bistable QS response in certain parameter regimes. We then combine the slow/fast analysis with a contraction mapping theorem in order to reduce a population model to an effective single-cell model, and show how the weight control mechanism allows bacteria to have a finer discrimination of their social and physical environment.

Bidirectional transport model of morphogen gradient formation via cytonemes.

Morphogen protein gradients play an important role in the spatial regulation of patterning during embryonic development. The most commonly accepted mechanism for gradient formation is diffusion from a source combined with degradation. Recently, there has been growing interest in an alternative mechanism, which is based on the direct delivery of morphogens along thin, actin-rich cellular extensions known as cytonemes. In this paper, we develop a bidirectional motor transport model for the flux of morphogens along cytonemes, linking a source cell to a one-dimensional array of target cells. By solving the steady-state transport equations, we show how a morphogen gradient can be established, and explore how the mean velocity of the motors affects properties of the morphogen gradient such as accumulation time and robustness. In particular, our analysis suggests that in order to achieve robustness with respect to changes in the rate of synthesis of morphogen, the mean velocity has to be negative, that is, retrograde flow or treadmilling dominates. Thus the potential targeting precision of cytonemes comes at an energy cost. We then study the effects of non-uniformly allocating morphogens to the various cytonemes projecting from a source cell. This competition for resources provides a potential regulatory control mechanism not available in diffusion-based models.

A variational method for analyzing limit cycle oscillations in stochastic hybrid systems.

Many systems in biology can be modeled through ordinary differential equations, which are piece-wise continuous, and switch between different states according to a Markov jump process known as a stochastic hybrid system or piecewise deterministic Markov process (PDMP). In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper, we develop a phase reduction method for stochastic hybrid systems that support a stable limit cycle in the deterministic limit. A classic example is the Morris-Lecar model of a neuron, where the switching Markov process is the number of open ion channels and the continuous process is the membrane voltage. We outline a variational principle for the phase reduction, yielding an exact analytic expression for the resulting phase dynamics. We demonstrate that this decomposition is accurate over timescales that are exponential in the switching rate ϵ-1. That is, we show that for a constant C, the probability that the expected time to leave an O(a) neighborhood of the limit cycle is less than T scales as T exp (-Ca/ϵ).

Synchronization of stochastic hybrid oscillators driven by a common switching environment.

Many systems in biology, physics, and chemistry can be modeled through ordinary differential equations (ODEs), which are piecewise smooth, but switch between different states according to a Markov jump process. In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper, we suppose that this limit ODE supports a stable limit cycle. We demonstrate that a set of such oscillators can synchronize when they are uncoupled, but they share the same switching Markov jump process. The latter is taken to represent the effect of a common randomly switching environment. We determine the leading order of the Lyapunov coefficient governing the rate of decay of the phase difference in the fast switching limit. The analysis bears some similarities to the classical analysis of synchronization of stochastic oscillators subject to common white noise. However, the discrete nature of the Markov jump process raises some difficulties: in fact, we find that the Lyapunov coefficient from the quasi-steady-state approximation differs from the Lyapunov coefficient one obtains from a second order perturbation expansion in the waiting time between jumps. Finally, we demonstrate synchronization numerically in the radial isochron clock model and show that the latter Lyapunov exponent is more accurate.

Isolation of rare recombinants without using selectable markers for one-step seamless BAC mutagenesis.

Current methods to isolate rare (1:10,000-1:100,000) bacterial artificial chromosome (BAC) recombinants require selectable markers. For seamless BAC mutagenesis, selectable markers need to be removed after isolation of recombinants through counterselection. Here we illustrate founder principle-driven enrichment (FPE), a simple method to rapidly isolate rare recombinants without using selectable markers, allowing one-step seamless BAC mutagenesis. As proof of principle, we isolated 1:100,000 seamless fluorescent protein-modified Nodal BACs and confirmed BAC functionality by generating fluorescent reporter mice. We also isolated small indel P1 phage-derived artificial chromosome (PAC) and BAC recombinants. Statistical analysis revealed that 1:100,000 recombinants can be isolated with <40 PCRs, and we developed a web-based calculator to optimize FPE.

Population Model of Quorum Sensing with Multiple Parallel Pathways.

Quorum sensing (QS) is a bacterial communication mechanism that uses signal-receptor binding to regulate gene expression based on cell density, resulting in group behaviors such as biofilm formation, bioluminescence and stress response. In certain bacterial species such as Vibrio harveyi, several parallel QS signaling pathways drive a single phosphorylation-dephosphorylation cycle, which in turn regulates QS target genes. In this paper, we investigate the possible role of parallel signaling pathways by developing a mathematical model of QS in V. harveyi at both the single-cell and population levels. First we explore how signal integration may be achieved at the single-cell level, and how different model parameters influence the process. We then consider two examples of signal integration at the population level: a one-population model responding to two environmental cues (cell density and mass transfer), and a two-population model with distinct cell densities. In each case, we use contraction analysis to reduce the population model to an effective single-cell model.

Laminar Neural Field Model of Laterally Propagating Waves of Orientation Selectivity.

We construct a laminar neural-field model of primary visual cortex (V1) consisting of a superficial layer of neurons that encode the spatial location and orientation of a local visual stimulus coupled to a deep layer of neurons that only encode spatial location. The spatially-structured connections in the deep layer support the propagation of a traveling front, which then drives propagating orientation-dependent activity in the superficial layer. Using a combination of mathematical analysis and numerical simulations, we establish that the existence of a coherent orientation-selective wave relies on the presence of weak, long-range connections in the superficial layer that couple cells of similar orientation preference. Moreover, the wave persists in the presence of feedback from the superficial layer to the deep layer. Our results are consistent with recent experimental studies that indicate that deep and superficial layers work in tandem to determine the patterns of cortical activity observed in vivo.

Model of Growth Cone Membrane Polarization via Microtubule Length Regulation.

We present a mathematical model of membrane polarization in growth cones. We proceed by coupling an active transport model of cytosolic proteins along a two-dimensional microtubule (MT) network with a modified Dogterom-Leibler model of MT growth. In particular, we consider a Rac1-stathmin-MT pathway in which the growth and catastrophe rates of MTs are regulated by cytosolic stathmin, while the stathmin is regulated by Rac1 at the membrane. We use regular perturbation theory and numerical simulations to determine the steady-state stathmin concentration, the mean MT length distribution, and the resulting distribution of membrane-bound proteins. We thus show how a nonuniform Rac1 distribution on the membrane generates a polarized distribution of membrane proteins. The mean MT length distribution and hence the degree of membrane polarization are sensitive to the precise form of the Rac1 distribution and parameters such as the catastrophe-promoting constant and tubulin association rate. This is a consequence of the fact that the lateral diffusion of stathmin tends to weaken the effects of Rac1 on the distribution of mean MT lengths.

Delayed feedback model of axonal length sensing.

A fundamental question in cell biology is how the sizes of cells and organelles are regulated at various stages of development. Size homeostasis is particularly challenging for neurons, whose axons can extend from hundreds of microns to meters (in humans). Recently, a molecular-motor-based mechanism for axonal length sensing has been proposed, in which axonal length is encoded by the frequency of an oscillating retrograde signal. In this article, we develop a mathematical model of this length-sensing mechanism in which advection-diffusion equations for bidirectional motor transport are coupled to a chemical signaling network. We show that chemical oscillations emerge due to delayed negative feedback via a Hopf bifurcation, resulting in a frequency that is a monotonically decreasing function of axonal length. Knockdown of either kinesin or dynein causes an increase in the oscillation frequency, suggesting that the length-sensing mechanism would produce longer axons, which is consistent with experimental findings. One major prediction of the model is that fluctuations in the transport of molecular motors lead to a reduction in the reliability of the frequency-encoding mechanism for long axons.

Synaptic democracy and vesicular transport in axons.

Synaptic democracy concerns the general problem of how regions of an axon or dendrite far from the cell body (soma) of a neuron can play an effective role in neuronal function. For example, stimulated synapses far from the soma are unlikely to influence the firing of a neuron unless some sort of active dendritic processing occurs. Analogously, the motor-driven transport of newly synthesized proteins from the soma to presynaptic targets along the axon tends to favor the delivery of resources to proximal synapses. Both of these phenomena reflect fundamental limitations of transport processes based on a localized source. In this Letter, we show that a more democratic distribution of proteins along an axon can be achieved by making the transport process less efficient. This involves two components: bidirectional or "stop-and-go" motor transport (which can be modeled in terms of advection-diffusion), and reversible interactions between motor-cargo complexes and synaptic targets. Both of these features have recently been observed experimentally. Our model suggests that, just as in human societies, there needs to be a balance between "efficiency" and "equality".

Entorhinal stellate cells show preferred spike phase-locking to theta inputs that is enhanced by correlations in synaptic activity.

In active networks, excitatory and inhibitory synaptic inputs generate membrane voltage fluctuations that drive spike activity in a probabilistic manner. Despite this, some cells in vivo show a strong propensity to precisely lock to the local field potential and maintain a specific spike-phase relationship relative to other cells. In recordings from rat medial entorhinal cortical stellate cells, we measured spike phase-locking in response to sinusoidal "test" inputs in the presence of different forms of background membrane voltage fluctuations, generated via dynamic clamp. We find that stellate cells show strong and robust spike phase-locking to theta (4-12 Hz) inputs. This response occurs under a wide variety of background membrane voltage fluctuation conditions that include a substantial increase in overall membrane conductance. Furthermore, the IH current present in stellate cells is critical to the enhanced spike phase-locking response at theta. Finally, we show that correlations between inhibitory and excitatory conductance fluctuations, which can arise through feedback and feedforward inhibition, can substantially enhance the spike phase-locking response. The enhancement in locking is a result of a selective reduction in the size of low-frequency membrane voltage fluctuations due to cancellation of inhibitory and excitatory current fluctuations with correlations. Hence, our results demonstrate that stellate cells have a strong preference for spike phase-locking to theta band inputs and that the absolute magnitude of locking to theta can be modulated by the properties of background membrane voltage fluctuations.

Stochastic hybrid model of spontaneous dendritic NMDA spikes.

Following recent advances in imaging techniques and methods of dendritic stimulation, active voltage spikes have been observed in thin dendritic branches of excitatory pyramidal neurons, where the majority of synapses occur. The generation of these dendritic spikes involves both Na(+) ion channels and M-methyl-D-aspartate receptor (NMDAR) channels. During strong stimulation of a thin dendrite, the resulting high levels of glutamate, the main excitatory neurotransmitter in the central nervous system and an NMDA agonist, modify the current-voltage (I-V) characteristics of an NMDAR so that it behaves like a voltage-gated Na(+) channel. Hence, the NMDARs can fire a regenerative dendritic spike, just as Na(+) channels support the initiation of an action potential following membrane depolarization. However, the duration of the dendritic spike is of the order 100 ms rather than 1 ms, since it involves slow unbinding of glutamate from NMDARs rather than activation of hyperpolarizing K(+) channels. It has been suggested that dendritic NMDA spikes may play an important role in dendritic computations and provide a cellular substrate for short-term memory. In this paper, we consider a stochastic, conductance-based model of dendritic NMDA spikes, in which the noise originates from the stochastic opening and closing of a finite number of Na(+) and NMDA receptor ion channels. The resulting model takes the form of a stochastic hybrid system, in which membrane voltage evolves according to a piecewise deterministic dynamics that is coupled to a jump Markov process describing the opening and closing of the ion channels. We formulate the noise-induced initiation and termination of a dendritic spike in terms of a first-passage time problem, under the assumption that glutamate unbinding is negligible, which we then solve using a combination of WKB methods and singular perturbation theory. Using a stochastic phase-plane analysis we then extend our analysis to take proper account of the combined effects of glutamate unbinding and noise on the termination of a spike.

Truncated stochastically switching processes.

There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a finite-state Markov chain, and the continuous dynamics depends on the current internal state. The resulting hybrid stochastic differential equation (hSDE) could describe the evolution of a neuron's membrane potential, the concentration of proteins synthesized by a gene network, or the position of an active particle. Another major class of switching system is a search process with stochastic resetting, where the position of a diffusing or active particle is reset to a fixed position at a random sequence of times. In this case the system switches between a search phase and a reset phase, where the latter may be instantaneous. In this paper, we investigate how the behavior of a stochastically switching system is modified when the maximum number of switching (or reset) events in a given time interval is fixed. This is motivated by the idea that each time the system switches there is an additive energy cost. We first show that in the case of an hSDE, restricting the number of switching events is equivalent to truncating a Volterra series expansion of the particle propagator. Such a truncation significantly modifies the moments of the resulting renormalized propagator. We then investigate how restricting the number of reset events affects the diffusive search for an absorbing target. In particular, truncating a Volterra series expansion of the survival probability, we calculate the splitting probabilities and conditional MFPTs for the particle to be absorbed by the target or exceed a given number of resets, respectively.