A Minimal Model of a Central Pattern Generator and Motoneurons for Insect Locomotion

Abstract

Abstract. We adapt the generic three-dimensional bursting neuron model derived in the companion paper [SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 636-670] to model central pattern generator interneurons and slow and fast motoneurons in insect locomotory systems. Focusing on cockroach data, we construct a coupled network that retains sufficient detail to allow investigation and prediction of biophysical parameter changes. We show that the model can encompass stepping frequency, duty cycle, and motoneuron output variations observed in cockroaches, and we reduce it to an analytically tractable symmetric network of coupled phase oscillators from which general principles can be extracted. The model's modular form allows dynamical analyses of individual components and the addition of other components, so we expect it to be more generally useful. 1. Introduction. Central pattern generators (CPGs) are networks of functionally distinguishable neurons, located in the vertebrate spinal cord or in invertebrate thoracic ganglia, capable of generating and regulating the spatio-temporal activity of motoneurons in the absence of sensory input (e.g., [2,3,4]). Over forty years of in vitro and in vivo studies of network architectures, intrinsic membrane properties, and neuromodulators (e.g., [5,6,7,4,8]) have firmly established their importance in motor behavior. CPG dynamics depends on intracellular, synaptic, and network level phenomena and can display remarkable richness and flexibility.In this paper, using the reduced bursting neuron ODEs derived and studied in the preceding paper [1], we develop a model of the CPG and associated bursting motoneurons for insect locomotion. We draw on data from the death's head and American cockroaches Blaberus discoidalis and Periplaneta americana and focus on rapid running, a regime in which preflexive feedforward control [9, 10] appears to dominate and reflexive feedback plays a less important role [11,12,13] than in, e.g., stick insects [14] that use more varied gaits and leg placement strategies. We include enough ionic current and conductance detail to reveal how modulation

Minimal Models of Bursting Neurons: How Multiple Currents, Conductances, and Timescales Affect Bifurcation Diagrams

“…We have two main goals: to integrate and extend a body of work, largely in theoretical and mathematical neuroscience, that enables (semi-) analytical studies of bursting neurons, while maintaining sufficient biophysical detail for comparisons with experimental data; and to use this to derive a model of a CPG that reveals how key locomotive properties may be determined by individual neurons and the network as a whole. In this first paper we show how complex models can be reduced and develop the analytical methods; in [1] we construct the CPG model.…”

“…Some network studies have also recently been done [17,26]. When limited experimental data is available, as in [17] and the CPG model of [1], generic models and broad parameter variations can still lead to testable hypotheses and provide motivation to verify novel predictions [27]. However, while asymptotic reductions and polynomial approximations aid mathematical analyses, they often obscure biophysical effects that must be retained if one is to understand how internal components and architecture, as well as proprioceptive sensing and commands from higher centers, can influence a network [28].…”

“…We have two main goals: to integrate and extend a body of work, largely in theoretical and mathematical neuroscience, that enables (semi-) analytical studies of bursting neurons, while maintaining sufficient biophysical detail for comparisons with experimental data; and to use this to derive a model of a CPG that reveals how key locomotive properties may be determined by individual neurons and the network as a whole. In this first paper we show how complex models can be reduced and develop the analytical methods; in [1] we construct the CPG model.The first dynamical neural model based on biophysical data was due to Hodgkin and Huxley [2], and their description of the action potential (AP) and ionic currents in the giant axon of the squid has been vastly extended and generalized in the half century since. Detailed axonal and dendritic geometry can be included, for example, at the unicellular level.…”

“…These models are specific to particular animals and even to in vitro preparations and, not being amenable to analytical comparative studies, do not readily reveal general principles. In this paper and [1] we seek a balance between such complexity and simpler phenomenological models employing phase oscillators [9,10] or connectionist circuits [11] that have been used to study network connectivity effects.…”

Towards Testable Neuromechanical Control Architectures for Running

“…Physiologically meaningful dynamical models of neurons [40] can be reduced to two [23,56] or three [30] dimensional dynamical systems in principled ways that retain the salient physiological dependencies with very few lumped parameters. In turn, these can be assembled as physiologically representative [59] modules, in a network of coupled oscillators that admits further mathematically principled reduction in dimension via phase variables [29]. The musculo-skeletal system is represented by mass-spring systems or second order oscillators.…”