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83
Type I Membranes, Phase Resetting Curves, and Synchrony
 Neural Comput
, 1995
"... Type I membrane oscillators such as the Connor model (Connor, Walter, and McKown, 1977) and the MorrisLecar 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 wit ..."
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Cited by 126 (13 self)
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Type I membrane oscillators such as the Connor model (Connor, Walter, and McKown, 1977) and the MorrisLecar 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 ...
Genetic Programming Approach to the Construction of a Neural Network for Control of a Walking Robot
 In IEEE International Conference on Robotics and Automation
, 1992
"... 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 algorit ..."
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Cited by 43 (2 self)
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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 illposed 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...
Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity
 Neural Computation
, 2003
"... In model networks of Ecells and Icells (excitatory and inhibitory neurons) , synchronous rhythmic spiking often comes about from the interplay between the two cell groups: the Ecells synchronize the Icells and vice versa. Under ideal conditions  homogeneity in relevant network parameters, ..."
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Cited by 42 (8 self)
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In model networks of Ecells and Icells (excitatory and inhibitory neurons) , synchronous rhythmic spiking often comes about from the interplay between the two cell groups: the Ecells synchronize the Icells and vice versa. Under ideal conditions  homogeneity in relevant network parameters, and alltoall connectivity for instance  this mechanism can yield perfect synchronization.
Dynamics of Membrane Excitability Determine Interspike Interval Variability: A Link Between Spike Generation Mechanisms and Cortical Spike Train Statistics
, 1998
"... 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 saddlenode dynamics underlie excitability (Rinzel & Ermentrout, 19 ..."
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Cited by 37 (4 self)
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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 saddlenode dynamics underlie excitability (Rinzel & Ermentrout, 1989). We present a canonical model for type I membranes, the θneuron. The θneuron 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 MorrisLecar (ML) neural model with random synaptic inputs: type I ML 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.
Ion Channel Stochasticity May Be Critical in Determining the Reliability and Precision of Spike Timing
, 1998
"... 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 zon ..."
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Cited by 30 (5 self)
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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 shortterm 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 activitydependent longterm changes in the properties of intrinsic membrane currents.
Decoding Neuronal Firing And Modeling Neural Networks
 Quart. Rev. Biophys
, 1994
"... 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 w ..."
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Cited by 25 (4 self)
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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; HarrisWarrick & 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
Geometric Singular Perturbation Analysis of Neuronal Dynamics
 in Handbook of Dynamical Systems
, 2000
"... : In this chapter, we consider recent models for neuronal activity. We review the sorts of oscillatory behavior which may arise from the models and then discuss how geometric singular perturbation methods have been used to analyze these rhythms. We begin by discussing models for single cells which d ..."
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Cited by 23 (13 self)
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: In this chapter, we consider recent models for neuronal activity. We review the sorts of oscillatory behavior which may arise from the models and then discuss how geometric singular perturbation methods have been used to analyze these rhythms. We begin by discussing models for single cells which display bursting oscillations. There are, in fact, several dierent classes of bursting solutions; these have been classied by the geometric properties of how solutions evolve in phase space. We describe several of the bursting classes and then review related rigorous mathematical analysis. We then discuss the dynamics of small networks of neurons. We are primarily interested in whether excitatory or inhibitory synaptic coupling leads to either synchronous or desynchronous rhythms. We demonstrate that all four combinations are possible, depending on the details of the intrinsic and synaptic properties of the cells. Finally, we discuss larger networks of neuronal oscillators involving two dis...
Dynamically detuned oscillations account for the coupled rate and temporal code of place cell firing. Hippocampus 13:700–714
"... ABSTRACT: Firing of place cells in the exploring rat conveys doubly coded spatial information: both the rate of spikes and their timing relative to the phase of the ongoing field theta oscillation are correlated with the location of the animal. Specifically, the firing rate of a place cell waxes and ..."
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Cited by 22 (4 self)
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ABSTRACT: Firing of place cells in the exploring rat conveys doubly coded spatial information: both the rate of spikes and their timing relative to the phase of the ongoing field theta oscillation are correlated with the location of the animal. Specifically, the firing rate of a place cell waxes and wanes, while the timing of spikes precesses monotonically as the animal traverses the portion of the environment preferred by the cell. We propose a mechanism for the generation of this firing pattern that can be applied for place cells in all three hippocampal subfields and that encodes spatial information in the output of the cell without relying on topographical connections or topographical input. A single pyramidal cell was modeled so that the cell received rhythmic inhibition in phase with theta field potential oscillation on the soma and was excited on the dendrite with input depending on the speed of the rat. The dendrite sustained an intrinsic membrane potential oscillation, frequency modulated by its input. Firing probability of the cell was determined jointly by somatic and
Mechanisms of PhaseLocking and Frequency Control in Pairs of coupled Neural Oscillators
, 2000
"... 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 associ ..."
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Cited by 21 (5 self)
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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