A minimal synaptic architecture is proposed for how the brain might perform path integration by computing the next internal representation of self-location from the current representation and from the perceived velocity of motion. In the model, a place-cell assembly called a “chart ” contains a twodimensional attractor set called an “attractor map ” that can be used to represent coordinates in any arbitrary environment, once associative binding has occurred between chart locations and sensory inputs. In hippocampus, there are different spatial relations among place fields in different environments and behavioral contexts. Thus, the same units may participate in many charts, and it is shown that the number of uncorrelated charts that can be encoded in the same recurrent network is potentially quite large. According to this theory, the firing of a given place cell is primarily a cooperative effect of the activity of its

this paper I present a network model of spiking neurons in which synapses are endowed with realistic gating kinetics, based on experimentally measured dynamical properties of cortical synapses. I will focus on how delay-period activity could be generated by neuronally plausible mechanisms; the issue of memory field formation will be addressed in a separate study. A main problem to be investigated is that of "rate control" for a persistent state: if a robust persistent activity necessitates strong recurrent excitatory connections, how can the network be prevented from runaway excitation in spite of the powerful positive feedback, so that neuronal firing rates are low and comparable to those of PFC cells (10 --50 Hz)? Moreover, a persistent state may be destabilized because of network dynamics. For example, fast recurrent excitation followed by a slower negative feedback may lead to network instability and a collapse of the persistent state. It is shown that persistent states at low firing rates are usually stable only in the presence of sufficiently slow excitatory synapses of the NMDA type. Functional implications of these results for the role of Received April 14, 1999; revised Aug. 12, 1999; accepted Aug. 12, 1999

Can one develop an abstract description of the dynamics of pattern generators that provides quantitative insight into their operation? We explored this question by examining the dynamics of a model central pattern generator that was created using an evolutionary algorithm. We propose an abstract description based on the concept of a dynamical module, a set of neurons that simultaneously make their transitions from one quasistable state to another while the synaptic inputs that they receive remain essentially constant, thus temporarily reducing the dimensionality of the circuit dynamics. Using the mathematical tools of dynamical systems theory, we describe a method for identifying dynamical modules, and demonstrate that this concept can be used to quantitatively characterize constraints on neural architecture, account for phase durations, and predict the effects of parameter changes. Moreover, this abstract description reveals coordinated parameter changes that leave the overall circuit...

Abstract. This article describes a large-scale model of turtle visual cortex that simulates the propagating waves of activity seen in real turtle cortex. The cortex model contains 744 multicompartment models of pyramidal cells, stellate cells, and horizontal cells. Input is provided by an array of 201 geniculate neurons modeled as single compartments with spike-generating mechanisms and axons modeled as delay lines. Diffuse retinal flashes or presentation of spots of light to the retina are simulated by activating groups of geniculate neurons. The model is limited in that it does not have a retina to provide realistic input to the geniculate, and the cortex and does not incorporate all of the biophysical details of real cortical neurons. However, the model does reproduce the fundamental features of planar propagating waves. Activation of geniculate neurons produces a wave of activity that originates at the rostrolateral pole of the cortex at the point where a high density of geniculate afferents enter the cortex. Waves propagate across the cortex with velocities of 4 µm/ms to 70 µm/ms and occasionally reflect from the caudolateral border of the cortex. Keywords: visual cortex, large-scale model, cortical waves, Karhunen-Loéve decomposition 1.

A procedure to calculate the Lyapunov characteristic exponent of the response of structural continuous systems, discretized using finite element methods, is proposed. The Lyapunov characteristic exponent can be used to characterize the asymptotic stability of the system dynamic response, and it is frequently employed to identify a chaotic behaviour. The proposed procedure can also be used in the stability characterization of fluid–structure interaction systems in which the focus of the analysis is on the

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Abstract. The result of computational operations performed at the single cell level are coded into sequences of action potentials (APs). In the cerebral cortex, due to its columnar organization, large number of neurons are involved in any individual processing task. It is therefore important to understand how the properties of coding at the level of neuronal populations are determined by the dynamics of single neuron AP generation. Here, we analyze how the AP generating mechanism determines the speed with which an ensemble of neurons can represent transient stochastic input signals. We analyze a generalization of the θ-neuron, the normal form of the dynamics of Type-I excitable membranes. Using a novel sparse matrix representation of the Fokker-Planck equation, which describes the ensemble dynamics, we calculate the transmission functions for small modulations of the mean current and noise noise amplitude. In the high-frequency limit the transmission function decays as ω −γ, where γ surprisingly depends on the phase θs at which APs are emitted. If at θs the dynamics is insensitive to external inputs, the transmission function decays as (i) ω −3 for the case of a modulation of a white noise input and as (ii) ω −2 for a modulation of the mean input current in the presence of a correlated and uncorrelated noise as well as (iii) in the case of a modulated amplitude of a correlated noise input. If the insensitivity condition is lifted, the transmission function always decays as ω −1,asinconductance based neuron models. In a physiologically plausible regime up to 1 kHz the typical response speed is, however, independent of the high-frequency limit and is set by the rapidness of the

The goal of the tutorial is to introduce Systems Biologists to the application of control theory to biology. The tools of systems and control theory have been instrumental in the successful design of countless man-made complex systems. Biologists, who are trying to reverse engineer complex living networks, can benefit from the insights this theory can provide into the design of biological systems. Of particular importance is understanding the role of feedback control in ensuring the robust behavior of biological processes subjected to internal and external disturbances. The intended audience is the general Systems Biology community. We will not assume technical knowledge beyond what a typical biologist would be exposed to as an undergraduate and graduate student. Relevant biological examples will be used to give a feel for more abstract concepts, followed up by mathematical explanations that are presented in detail in the handouts. Prerequisites: understanding of chemical and enzyme kinetics, familiarity with ordinary differential equations and linear algebra. Tutorial Content The tutorial will begin with a discussion of the connections between control theory and Metabolic Control Analysis (MCA). This section will serve as an introduction to sensitivity analysis and some of the basic tools of control theory. The next section will cover a specific control structure which plays

Abstract: A system dynamics analysis of consciousness during a ten-hour religious experience of purgation, which just preceded an estimated four-to-seven second experience of mystical union, gives purgation a scientific structure. Using that structure, my research in comparative religion shows that at the essence or core of each religion resides this same sacred structure and essence. As a result, the analysis forms a general theory of religion: e pluribus unum. The work then goes further, establishing the long-sought-for integration of science and religion by rooting this core of religion in biology. As a byproduct, the analysis solves the central problem in the emerging field of consciousness studies, Chalmers ' hard problem. The presentation contains much personal material on my religious life and experiences. This is necessary as preliminary for the formalized analyses of such material. The fullest presentation of these ideas- with over 400 links- is at:

Third-order explicit autonomous differential equations, commonly called jerky dynamics, constitute a powerful approach to understand the properties of functionally very simple but nonlinear three-dimensional dynamical systems that can exhibit chaotic longtime behavior. In this paper, we investigate the dynamics that can be generated by the two simplest polynomial jerky dynamics that, up to these days, are known to show chaotic behavior in some parameter range. After deriving several analytical properties of these systems, we systematically determine the dependence of the long-time dynamical behavior on the system parameters by numerical evaluation of Lyapunov spectra. Some features of the systems that are related to the dependence on initial conditions are also addressed. The observed dynamical complexity of the two systems is discussed in connection with the existence of homoclinic orbits.