Bart Sautois

New features of the software MatCont for bifurcation analysis of dynamical systems

Bifurcation software is an essential tool in the study of dynamical systems. From the beginning (the first packages were written in the 1970s) 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. Poincaré 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.

Reconfiguration of a Vertebrate Motor Network: Specific Neuron Recruitment and Context-Dependent Synaptic Plasticity

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 Browns (1911) hypothesis for locomotor rhythmogenesis.

Computation of the Phase Response Curve: A Direct Numerical Approach

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.

Axon and dendrite geography predict the specificity of synaptic connections in a functioning spinal cord network

BackgroundHow 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.ResultsThe probabilities of synapses between 7 types of identified spinal neuron were measured directly by making electrical recordings from 500 pairs of neurons. For the same neuron types, the dorso-ventral distributions of axons and dendrites were measured and then used to calculate the probabilities that axons would encounter particular dendrites and so potentially form synaptic connections. Surprisingly, synapses were found between all types of neuron but contact probabilities could be predicted simply by the anatomical overlap of their axons and dendrites. These results suggested that synapse formation may not require axons to recognise specific, correct dendrites. To test the plausibility of simpler hypotheses, we first made computational models that were able to generate longitudinal axon growth paths and reproduce the axon distribution patterns and synaptic contact probabilities found in the spinal cord. To test if probabilistic rules could produce functioning spinal networks, we then made realistic computational models of spinal cord neurons, giving them established cell-specific properties and connecting them into networks using the contact probabilities we had determined. A majority of these networks produced robust swimming activity.ConclusionSimple factors such as morphogen gradients controlling dorso-ventral soma, dendrite and axon positions may sufficiently constrain the synaptic connections made between different types of neuron as the spinal cord first develops and allow functional networks to form. Our analysis implies that detailed cellular recognition between spinal neuron types may not be necessary for the reliable formation of functional networks to generate early behaviour like swimming.

Role of type-specific neuron properties in a spinal cord motor network

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).

Continuation of homoclinic orbits in MATLAB

We have added the functionality for continuing homoclinic orbits to cl_matcont, a user-friendly matlab package for the study of dynamical systems and their bifurcations. It is now possible to continue homoclinic-to-hyperbolic-saddle and homoclinic-to-saddle-node orbits. The implementation is done using the continuation of invariant subspaces, with the Ricatti equations included in the defining system. The continuation can be initiated from a limit cycle with large period or from a Bogdanov-Takens point. All known codimension-two bifurcations are tested for, during continuation. The test functions for inclination-flip bifurcations are implemented in a new and more efficient way.

The onset and extinction of neural spiking: a numerical bifurcation approach.

We study the onset of neural spiking when the equilibrium rest state loses stability by the change of a critical parameter, the applied current. In the case of the well-known Morris-Lecar model, we start from a complete numerical study of the bifurcation diagram in the most relevant two-parameter range. This diagram includes all equilibrium and limit cycle bifurcations, thus correcting and completing earlier studies.We discuss and classify the behavior of the spiking orbits, when increasing or decreasing the applied current. A complete classification can be extracted from the complete bifurcation diagram. It is based on three components: bifurcation type of the equilibrium at the loss of stability, subcritical behavior in the limit of decreasing the applied current and supercritical behavior in the limit of increasing the applied current.

Bifurcation software in Matlab with applications in neuronal modeling

Many biological phenomena, notably in neuroscience, can be modeled by dynamical systems. We describe a recent improvement of a Matlab software package for dynamical systems with applications to modeling single neurons and all-to-all connected networks of neurons. The new software features consist of an object-oriented approach to bifurcation computations and the partial inclusion of C-code to speed up the computation. As an application, we study the origin of the spiking behaviour of neurons when the equilibrium state is destabilized by an incoming current. We show that Class II behaviour, i.e. firing with a finite frequency, is possible even if the destabilization occurs through a saddle-node bifurcation. Furthermore, we show that synchronization of an all-to-all connected network of such neurons with only excitatory connections is also possible in this case.

Switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in ODEs

The paper provides full algorithmic details on switching to the continuation of all possible codim 1 cycle bifurcations from generic codim 2 equilibrium bifurcation points in n-dimensional ODEs. We discuss the implementation and the performance of the algorithm in several examples, including an extended Lorenz-84 model and a laser system.

Phase response curves, delays and synchronization in MATLAB

MATCONT is a Matlab software package for the study of dynamical systems. We extended this package with the functionality of computing the phase response curve or PRC of a neural model. An important application lies in the study of synchronization of neurons. In synchronization studies, often the delays in a network are discarded or set to zero. We show in this paper that the delay can be crucial for the ability of neurons in an excitatory network to synchronize. We also show that, by studying the PRC of a neural model, one can compute the necessary delay to allow synchronization or phase-locking in a network of such neurons.