Bifurcation software is an essential tool in the study of dynamical systems. From the beginning (the first packages were written in the 1970's) it was also used in the modelling process, in particular to determine the values of critical parameters. More recently, it is used in a systematic way in the design of dynamical models and to determine which parameters are relevant. MATCONT and CL_MATCONT are freely available MATLAB numerical continuation packages for the interactive study of dynamical systems and bifurcations. MATCONT is the GUI-version, CL_MATCONT is the command-line version. The work started in 2000 and the first publications appeared in 2003. Since that time many new functionalities were added. Some of these are fairly simple but were never before implemented in continuation codes, e.g. Poincare´maps. Others were only available as toolboxes that can be used by experts, e.g. continuation of homoclinic orbits. Several others were never implemented at all, such as periodic normal forms for codimension 1 bifurcations of limit cycles, normal forms for codimension 2 bifurcations of equilibria, detection of codimension 2 bifurcations of limit cycles, automatic computation of phase response curves and their derivatives, continuation of branch points of equilibria and limit cycles. New numerical algorithms for these computations have been published or will appear elsewhere; here we restrict to their software implementation. We further discuss software issues that are in practice important for many users, e.g. how to define a new system starting from an existing one, how to import and export data, system descriptions, and computed results.

Motor networks typically generate several related output patterns or gaits where individual neurons may be shared or recruited between patterns. We investigate how a vertebrate locomotor network is reconfigured to produce a second rhythmic motor pattern, defining the detailed pattern of neuronal recruitment and consequent changes in the mechanism for rhythm generation. Hatchling Xenopus tadpoles swim if touched, but when held make slower, stronger, struggling movements. In immobilized tadpoles, a brief current pulse to the skin initiates swimming, whereas 40 Hz pulses produce struggling. The classes of neurons active during struggling are defined using whole-cell patch recordings from hindbrain and spinal cord neurons during 40 Hz stimulation of the skin. Some motoneurons and inhibitory interneurons are active in both swimming and struggling, but more neurons are recruited within these classes during struggling. In addition, and in contrast to a previous study, we describe two new classes of excitatory interneuron specifically recruited during struggling and define their properties and synaptic connections. We then explore mechanisms that generate struggling by building a network model incorporating these new neurons. As well as the recruitment of new neuron classes, we show that reconfiguration of the locomotor network to the struggling central pattern generator (CPG) reveals a context-dependent synaptic depression of reciprocal inhibition: the result of increased inhibitory neuron firing frequency during struggling. This provides one possible mechanism for burst termination not seen in the swimming CPG. The direct demonstration of depression in reciprocal inhibition confirms a key element of Brown's (1911) hypothesis for locomotor rhythmogenesis.

Recent recordings from spinal neurons in hatchling frog tadpoles allow their type-specific properties to be defined. Seven main types of neuron involved in the control of swimming have been characterized. To investigate the significance of type-specific properties, we build models of each neuron type and assemble them into a network using known connectivity between: sensory neurons, sensory pathway interneurons, central pattern generator (CPG) interneurons and motoneurons. A single stimulus to a sensory neuron initiates swimming where modelled neuronal and network activity parallels physiological activity. Substitution of firing properties between neuron types shows that those of excitatory CPG interneurons are critical for stable swimming. We suggest that type-specific neuronal properties can reflect the requirements for involvement in one particular network response (like swimming), but may also reflect the need to participate in more than one response (like swimming and slower struggling).

Background: How specific are the synaptic connections formed as neuronal networks develop and can simple rules account for the formation of functioning circuits? These questions are assessed in the spinal circuits controlling swimming in hatchling frog tadpoles. This is possible because detailed information is now available on the identity and synaptic connections of the main types of neuron.

Neurons are often modeled by dynamical systems--parameterized systems of differential equations. A typical behavioral pattern of neurons is periodic spiking; this corresponds to the presence of stable limit cycles in the dynamical systems model. The phase resetting and phase response curves (PRCs) describe the reaction of the spiking neuron to an input pulse at each point of the cycle. We develop a new method for computing these curves as a by-product of the solution of the boundary value problem for the stable limit cycle. The method is mathematically equivalent to the adjoint method, but our implementation is computationally much faster and more robust than any existing method. In fact, it can compute PRCs even where the limit cycle can hardly be found by time integration, for example, because it is close to another stable limit cycle. In addition, we obtain the discretized phase response curve in a form that is ideally suited for most applications. We present several examples and provide the implementation in a freely available Matlab code.