New patterns of activity in a pair of interacting excitatory-inhibitory neural fields.

In this Letter, we study stationary bump solutions in a pair of interacting excitatory-inhibitory (E-I) neural fields in one dimension. We demonstrate the existence of localized bump solutions of persistent activity that can be maintained by the pair of interacting layers when a stationary bump is not supported by either layer in isolation--a scenario which may be relevant as a mechanism for the persistent activity associated with working memory in the prefrontal cortex and may explain why bumps are not seen in in vitro slice preparations. Furthermore, we describe a new type of stationary bump solution arising from a pitchfork bifurcation which produces a stationary bump in each layer with a spatial offset that increases with the bifurcation parameter.

Traveling electrical waves in cortex: insights from phase dynamics and speculation on a computational role.

The theory of coupled phase oscillators provides a framework to understand the emergent properties of networks of neuronal oscillators. When the architecture of the network is dominated by short-range connections, the pattern of electrical output is predicted to correspond to traveling plane and rotating waves, in addition to synchronized output. We argue that this theory provides the foundation for understanding the traveling electrical waves that are observed across olfactory, visual, and visuomotor areas of cortex in a variety of species. The waves are typically present during periods outside of stimulation, while synchronous activity typically dominates in the presence of a strong stimulus. We suggest that the continuum of phase shifts during epochs with traveling waves provides a means to scan the incoming sensory stream for novel features. Experiments to test our theoretical approach are presented.

Modelling of intersegmental coordination in the lamprey central pattern generator for locomotion.

Rhythmic motor activity requires coordination of different muscles or muscle groups so that they are all active with the same cycle duration and appropriate phase relationships. The neural mechanisms for such phase coupling in vertebrate locomotion are not known. Swimming in the lamprey is accomplished by the generation of a travelling wave of body curvature in which the phase coupling between segments is so controlled as to give approximately one full wavelength on the body at any swimming speed. This article reviews work that has combined mathematical analysis, biological experimentation and computer simulation to provide a conceptual framework within which intersegmental coordination can be investigated. Evidence is provided to suggest that in the lamprey, ascending coupling is dominant over descending coupling and controls the intersegmental phase lag during locomotion. The significance of long-range intersegmental coupling is also discussed.

Computational study of NMDA conductance and cortical oscillations in schizophrenia.

N-methyl-D-aspartate (NMDA) receptor hypofunction has been implicated in the pathophysiology of schizophrenia. The illness is also characterized by gamma oscillatory disturbances, which can be evaluated with precise frequency specificity employing auditory cortical entrainment paradigms. This computational study investigates how synaptic NMDA hypofunction may give rise to network level oscillatory deficits as indexed by entrainment paradigms. We developed a computational model of a local cortical circuit with pyramidal cells and fast-spiking interneurons (FSI), incorporating NMDA, α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic (AMPA), and γ-aminobutyric acid (GABA) synaptic kinetics. We evaluated the effects of varying NMDA conductance on FSIs and pyramidal cells, as well as AMPA to NMDA ratio. We also examined the differential effects across a broad range of entrainment frequencies as a function of NMDA conductance. Varying NMDA conductance onto FSIs revealed an inverted-U relation with network gamma whereas NMDA conductance onto the pyramidal cells had a more monotonic relationship. Varying NMDA vs. AMPA conductance onto FSIs demonstrated the necessity of AMPA in the generation of gamma while NMDA receptors had a modulatory role. Finally, reducing NMDA conductance onto FSI and varying the stimulus input frequency reproduced the specific reductions in gamma range (~40 Hz) as observed in schizophrenia studies. Our computational study showed that reductions in NMDA conductance onto FSIs can reproduce similar disturbances in entrainment to periodic stimuli within the gamma range as reported in schizophrenia studies. These findings provide a mechanistic account of how specific cellular level disturbances can give rise to circuitry level pathophysiologic disturbance in schizophrenia.

Bistability in apoptosis: roles of bax, bcl-2, and mitochondrial permeability transition pores.

We propose a mathematical model for mitochondria-dependent apoptosis, in which kinetic cooperativity in formation of the apoptosome is a key element ensuring bistability. We examine the role of Bax and Bcl-2 synthesis and degradation rates, as well as the number of mitochondrial permeability transition pores (MPTPs), on the cell response to apoptotic stimuli. Our analysis suggests that cooperative apoptosome formation is a mechanism for inducing bistability, much more robust than that induced by other mechanisms, such as inhibition of caspase-3 by the inhibitor of apoptosis (IAP). Simulations predict a pathological state in which cells will exhibit a monostable cell survival if Bax degradation rate is above a threshold value, or if Bax expression rate is below a threshold value. Otherwise, cell death or survival occur depending on initial caspase-3 levels. We show that high expression rates of Bcl-2 can counteract the effects of Bax. Our simulations also demonstrate a monostable (pathological) apoptotic response if the number of MPTPs exceeds a threshold value. This study supports our contention, based on mathematical modeling, that cooperativity in apoptosome formation is critically important for determining the healthy responses to apoptotic stimuli, and helps define the roles of Bax, Bcl-2, and MPTP vis-à-vis apoptosome formation.

The behavior of rings of coupled oscillators.

Coupled oscillators in a ring are studied using perturbation and numerical methods. Stability of waves with nearest neighbor weak coupling is shown for a class of simple oscillators. Linkens' model for colorectal activity is analyzed and several stable modes are found. Stability of waves with general (non nearest neighbor coupling) is determined and comparisons to the nearest neighbor case are made. Approximate solutions to a ring with inhomogeneities are compared with numerical simulations.

A mathematical theory of visual hallucination patterns.

Neuronal activity in a two-dimensional net is analyzed in the neighborhood of an instability. Bifurcation theory and group theory are used to demonstrate the existence of a variety of doubly-periodic patterns, hexagons, rolls, etc., as solutions to the field equations for the net activity. It is suggested that these simple geometric patterns are the cortical concomitants of the "form constants" seen during visual hallucinosis.

Models for contact-mediated pattern formation: cells that form parallel arrays.

Kinetic continuum models are derived for cells that crawl over a 2D substrate, undergo random reorientation, and turn in response to contact with a neighbor. The integro-partial differential equations account for changes in the distribution of orientations in the population. It is found that behavior depends on parameters such as total mass, random motility, adherence, and sloughing rates, as well as on broad aspects of the contact response. Linear stability analysis, and numerical, and cellular automata simulations reveal that as parameters are varied, a bifurcation leads to loss of stability of a uniform (isotropic) steady state, in favor of an (anisotropic) patterned state in which cells are aligned in parallel arrays.

Model of the early development of thalamo-cortical connections and area patterning via signaling molecules.

The mammalian cortex is divided into architectonic and functionally distinct areas. There is growing experimental evidence that their emergence and development is controlled by both epigenetic and genetic factors. The latter were recently implicated as dominating the early cortical area specification. In this paper, we present a theoretical model that explicitly considers the genetic factors and that is able to explain several sets of experiments on cortical area regulation involving transcription factors Emx2 and Pax6, and fibroblast growth factor FGF8. The model consists of the dynamics of thalamo-cortical connections modulated by signaling molecules that are regulated genetically, and by axonal competition for neocortical space. The model can make predictions and provides a basic mathematical framework for the early development of the thalamo-cortical connections and area patterning that can be further refined as more experimental facts become known.

A model for actin-filament length distribution in a lamellipod.

A mathematical model is derived to describe the distributions of lengths of cytoskeletal actin filaments, along a 1 D transect of the lamellipod (or along the axis of a filopod) in an animal cell. We use the facts that actin filament barbed ends are aligned towards the cell membrane and that these ends grow rapidly in the presence of actin monomer as long as they are uncapped. Once a barbed end is capped, its filament tends to be degraded by fragmentation or depolymerization. Both the growth (by polymerization) and the fragmentation by actin-cutting agents are depicted in the model, which takes into account the dependence of cutting probability on the position along a filament. It is assumed that barbed ends are capped rapidly away from the cell membrane. The model consists of a system of discrete-integro-PDE's that describe the densities of barbed filament ends as a function of spatial position and length of their actin filament "tails". The population of capped barbed ends and their trailing filaments is similarly represented. This formulation allows us to investigate hypotheses about the fragmentation and polymerization of filaments in a caricature of the lamellipod and compare theoretical and observed actin density profiles.

Contact response of cells can mediate morphogenetic pattern formation.

Theories of morphogenetic pattern formation have included Turing's chemical prepatterns, mechanochemical interactions, cell sorting, and other mechanisms involving guided motion or signalling of cells. Many of these theories presuppose long-range cellular communication or other controls such as chemical concentration fields. However, the possibility that direct interactions between cells can lead to order and structure has not been seriously investigated in mathematical models. In this paper we consider this possibility, with emphasis on cells that reorient and align with each other when they come into contact. We show that such contact responses can account for the formation of multicellular patterns called parallel arrays. These patterns typically occur in tissue cultures of fibroblasts, and consist of clusters of cells sharing a common axis of orientation. Using predictions of a mathematical model and computer simulations of cell motion and interactions we show that contact responses alone, in the absence of other global controls, can promote the formation of these patterns. We suggest other situations in which patterns may result from direct cellular communication. Previous theories of morphogenesis are briefly reviewed and compared with this proposed mechanism.

Bistability in pulse propagation in networks of excitatory and inhibitory populations.

We study the propagation of traveling solitary pulses in one-dimensional networks of excitatory and inhibitory integrate-and-fire neurons. Slow pulses, during which inhibitory cells fire well before neighboring excitatory cells, can propagate along the network at intermediate inhibition levels. At higher levels, they destabilize via a Hopf bifurcation. There is a bistable parameter regime in which both fast and slow pulses can propagate. Lurching pulses with spatiotemporal periodicity can propagate in regimes for which continuous pulses do not exist.

Models for spatial polymerization dynamics of rod-like polymers.

We investigate the polymerization kinetics of rod-like polymer filaments interacting with a distribution of monomer in one spatial dimension (e.g. along a narrow tube). We consider a variety of possible cases, including competition by the filament tips for the available monomer, and behaviour analogous to "treadmilling" in which the polymer adds subunits to one end and loses them at the other end so as to maintain a constant length. Applications to biological polymers such as actin filaments and microtubules are discussed.

Dynamics of membrane excitability determine interspike interval variability: a link between spike generation mechanisms and cortical spike train statistics.

We propose a biophysical mechanism for the high interspike interval variability observed in cortical spike trains. The key lies in the nonlinear dynamics of cortical spike generation, which are consistent with type I membranes where saddle-node dynamics underlie excitability (Rinzel & Ermentrout, 1989). We present a canonical model for type I membranes, the theta-neuron. The theta-neuron is a phase model whose dynamics reflect salient features of type I membranes. This model generates spike trains with coefficient of variation (CV) above 0.6 when brought to firing by noisy inputs. This happens because the timing of spikes for a type I excitable cell is exquisitely sensitive to the amplitude of the suprathreshold stimulus pulses. A noisy input current, giving random amplitude "kicks" to the cell, evokes highly irregular firing across a wide range of firing rates; an intrinsically oscillating cell gives regular spike trains. We corroborate the results with simulations of the Morris-Lecar (M-L) neural model with random synaptic inputs: type I M-L yields high CVs. When this model is modified to have type II dynamics (periodicity arises via a Hopf bifurcation), however, it gives regular spike trains (CV below 0.3). Our results suggest that the high CV values such as those observed in cortical spike trains are an intrinsic characteristic of type I membranes driven to firing by "random" inputs. In contrast, neural oscillators or neurons exhibiting type II excitability should produce regular spike trains.

Continuous and lurching traveling pulses in neuronal networks with delay and spatially decaying connectivity.

Propagation of discharges in cortical and thalamic systems, which is used as a probe for examining network circuitry, is studied by constructing a one-dimensional model of integrate-and-fire neurons that are coupled by excitatory synapses with delay. Each neuron fires only one spike. The velocity and stability of propagating continuous pulses are calculated analytically. Above a certain critical value of the constant delay, these pulses lose stability. Instead, lurching pulses propagate with discontinuous and periodic spatio-temporal characteristics. The parameter regime for which lurching occurs is strongly affected by the footprint (connectivity) shape; bistability may occur with a square footprint shape but not with an exponential footprint shape. For strong synaptic coupling, the velocity of both continuous and lurching pulses increases logarithmically with the synaptic coupling strength g(syn) for an exponential footprint shape, and it is bounded for a step footprint shape. We conclude that the differences in velocity and shape between the front of thalamic spindle waves in vitro and cortical paroxysmal discharges stem from their different effective delay; in thalamic networks, large effective delay between inhibitory neurons arises from their effective interaction via the excitatory cells which display postinhibitory rebound.

Temporal oscillations in neuronal nets.

A model for the interactions of cortical neurons is derived and analyzed. It is shown that small amplitude spatially inhomogeneous standing oscillations can bifurcate from the rest state. In a periodic domain, traveling wave trains exist. Stability of these patterns is discussed in terms of biological parameters. Homoclinic and heteroclinic orbits are demonstrated for the space-clamped system.

Beyond a pacemaker's entrainment limit: phase walk-through.

Desynchronization occurs when a pacemaker is externally stimulated at too high a rate, beyond the entrainment limit. One may then observe phase walk-through: the pacemaker and stimulus phase difference, rather than lock to a fixed pattern, may cycle repeatedly through all values. By means of a simple but nonlinear one-variable model we describe qualitatively the properties of entrainment, loss of entrainment, and phase walk-through for rhythmically stimulated pacemakers. We obtain an expression for the cycle length (beat period) of the repetitive phase walk-through. This shows, in contrast to the simplest treatment, that in general beat frequency is not a simple linear function of the difference between the pacemaker and stimulus frequencies. Our results are illustrated in the context of a model experimental pacemaker, the rhythmically flashing firefly. We discuss how these modeling results apply in a much more general setting.

Cellular automata approaches to biological modeling.

We review a number of biologically motivated cellular automata (CA) that arise in models of excitable and oscillatory media, in developmental biology, in neurobiology, and in population biology. We suggest technical and theoretical arguments that permit greater speed and enhanced realism, and apply these to several classical examples of pattern formation. We also describe CA that arise in models for fibroblast aggregation, branching networks, trail following, and neuronal maps.

Models for the length distributions of actin filaments: I. Simple polymerization and fragmentation.

We studied mathematical models for the length distributions of actin filaments under the effects of polymerization/depolymerization, and fragmentation. In this paper, we emphasize the effects of these two processes acting alone. In this case, simple discrete and continuous models can be derived and solved explicitly (in several special cases), making the problem interesting from a modeling and pedagogical point of view. In a companion paper (Ermentrout and Edelstein-Keshet, 1998, Bull. Math. Biol. 60, 477-503) we investigate what happens when the processes act together, with particular attention to fragmentation by gelsolin, and with a greater level of biological detail.