Type I membrane oscillators such as the Connor model (Connor, Walter, and McKown, 1977) and the Morris-Lecar model (Morris and Lecar, 1981) admit very low frequency oscillations near the critical applied current. Hansel et.al., (1995) have numerically shown that synchrony is difficult to achieve with these models and that the phase resetting curve is strictly positive. We use singular perturbation methods and averaging to show that this is a general property of Type I membrane models. We show in a limited sense that so called type 2 resetting occurs with models that obtain rhythmicity via a Hopf bifurcation. We also show the differences between synapses that act rapidly and those that act slowly and derive a canonical form for the phase interactions. 1 Introduction The behavior of coupled neural oscillators has been the subject of a great deal of recent interest. In general, this behavior is quite difficult to analyze. Most of the results to date are primarily based on simulations of ...

This paper describes the staged evolution of a complex motor pattern generator (MPG) for the control of a walking robot. The MPG is composed of a network of neurons with weights determined by Genetic Algorithm (GA) optimization. Staged evolution is used to improve the convergence rate of the algorithm. First, an oscillator for the individual leg movements is evolved. Then, a network of these oscillators is evolved to coordinate the movements of the different legs. By introducing a staged set of manageable challenges, the algorithm's performance is improved. These techniques may be applicable to other complex or ill-posed control problems in robot control. 1 Introduction It is well known that intelligent robots must interact closely with the world. This interaction might be considered a discourse between the the computational structure of the robot and structure of world, mediated by sensor and actuators. The design of this computational structure is a formidable task. The engineer mu...

In model networks of E-cells and I-cells (excitatory and inhibitory neurons) , synchronous rhythmic spiking often comes about from the interplay between the two cell groups: the E-cells synchronize the I-cells and vice versa. Under ideal conditions --- homogeneity in relevant network parameters, and all-to-all connectivity for instance --- this mechanism can yield perfect synchronization.

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 θ-neuron. The θ-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.

This memory is embedded in the distribution of channel states in the spike initiation site. The nature and resolution of this memory depend on the size of the channel pool and on the kinetics and number of states of the channels. We hypothesize that the number of channels in the spike initiation zone may be optimized in some sense to give the reliability and accuracy discussed above, together with a short-term memory of the neuron's activity. In this context, it is interesting to mention the work of Marder, Abbott, Turrigiano, Liu, and Golowasch (1996) and Abbott et al. (1996), which demonstrates activity-dependent long-term changes in the properties of intrinsic membrane currents.

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We analyze the control of frequency for a synchronized inhibitory neuronal network. The analysis is done for a reduced membrane model with a biophysically based synaptic influence. We argue that such a reduced model can quantitatively capture the frequency behavior of a larger class of neuronal models. We show that in different parameter regimes, the network frequency depends in different ways on the intrinsic and synaptic time constants. Only in one portion of the parameter space, called phasic, is the network period proportional to the synaptic decay time. These results are discussed in connection with previous work of the authors, which showed that for mildly heterogeneous networks, the synchrony breaks down, but coherence is preserved much more for systems in the phasic regime than in the other regimes. These results imply that for mildly heterogeneous networks, the existence of a coherent rhythm implies a linear dependence of the network period on synaptic decay time and a much weaker dependence on the drive to the cells. We give experimental evidence for this conclusion.

Introduction Biological neural networks are large systems of complex elements interacting through a complex array of connections. Individual neurons express a large number of active conductances (Connors et al., 1982; Adams & Gavin, 1986; Llin'as, 1988; McCormick, 1990; Hille, 1992) and exhibit a wide variety of dynamic behaviors on time scales ranging from milliseconds to many minutes (Llin'as, 1988; Harris-Warrick & Marder, 1991; Churchland & Sejnowski, 1992; Turrigiano et al., 1994). Neurons in cortical circuits are typically coupled to thousands of other neurons (Stevens, 1989) and very little is known about the strengths of these synapses (although see Rosenmund et al., 1993; Hessler et al., 1993; Smetters & Nelson, 1993). The complex firing patterns of large neuronal populations are difficult to describe let alone understand. There is little point in accurately modeling each membrane potential in a large neural

this article (see Figure 1). By transforming equation 2.1 to the output space defined by o #) and setting the time derivative to 0, we find that the SSIO curve of a neuron with self-weight w is given by I # -1 (o) w o. A single additive model neuron can exhibit either unistable or bistable dynamics, depending on the strength of its selfweight and its net input (Cowan & Ermentrout, 1978). In a single CTRNN neuron, only unistable dynamics are possible when w<4 (see Figure 1A). When w>4, bistable dynamics occurs when I L (w) I R (w) (see -4-2 0 2 4 I+ 0 0.2 0.4 0.6 0.8 1 -4-2 0 2 4 I+ 0 0.2 0.4 0.6 0.8 1 Figure 1: Representative steady-state input-output (SSIO) diagrams of a single CTRNN for (A) w 2 and (B) w 8. The solid line shows the output space location of the neuron's equilibrium points as a function of the net input I # . Note that the SSIO becomes folded for w>4, indicating the existence of three equilibrium points. When the SSIO is folded, the left and right edges of the fold are given by I L (w)andI R (w), respectively (black points in B). The ranges of synaptic inputs received from other neurons are indicated by gray rectangles. The lower (min and upper (max limits of this range play an important role in the analysis described in this article. In both plots, two synaptic input ranges are shown: one for which the neuron is saturated off (left rectangle) and one for which the neuron is saturated on (right rectangle). The dashed line in A shows the piecewise linear SSIO approximation used in section 4.2, which suggests using the intersections of the linear pieces (black points) as the analog of the fold edges in part B

INTRODUCTION Oscillations occur in many networks of neurons, and are associated with motor behavior, sensory processing, learning, arousal, attention and pathology (Parkinson's tremor, epilepsy). Such oscillations can be generated in many ways. This chapter discusses some mathematical issues associated with creation of coherent rhythmic activity in networks of neurons. We focus on pairs of cells, since many of the issues for larger networks are most clearly displayed in that context. As we will show, there are many mechanisms for interactions among the network components, and these can have different mathematical properties. A description of behavior of larger networks using some of the mechanisms described in this chapter can be found in the related chapter by Rubin and Terman. For reviews of papers about oscillatory behavior in specific networks in the nervous system, see Gray (1994), Marder and Calabrese (1996), Singer (1993), and Traub et al (1999). The chapter is organiz