A Salamander’s Flexible Spinal Network for Locomotion, Modeled at Two Levels of Abstraction

Animals have to coordinate a large number of muscles in different ways to efficiently move at various speeds and in different and complex environments. This coordination is in large part based on central pattern generators (CPGs). These neural networks are capable of producing complex rhythmic patterns when activated and modulated by relatively simple control signals. Although the generation of particular gaits by CPGs has been successfully modeled at many levels of abstraction, the principles underlying the generation and selection of a diversity of patterns of coordination in a single neural network are still not well understood. The present work specifically addresses the flexibility of the spinal locomotor networks in salamanders. We compare an abstract oscillator model and a CPG network composed of integrate-and-fire neurons, according to their ability to account for different axial patterns of coordination, and in particular the transition in gait between swimming and stepping modes. The topology of the network is inspired by models of the lamprey CPG, complemented by additions based on experimental data from isolated spinal cords of salamanders. Oscillatory centers of the limbs are included in a way that preserves the flexibility of the axial network. Similarly to the selection of forward and backward swimming in lamprey models via different excitation to the first axial segment, we can account for the modification of the axial coordination pattern between swimming and forward stepping on land in the salamander model, via different uncoupled frequencies in limb versus axial oscillators (for the same level of excitation). These results transfer partially to a more realistic model based on formal spiking neurons, and we discuss the difference between the abstract oscillator model and the model built with formal spiking neurons.

From lamprey to salamander: an exploratory modeling study on the architecture of the spinal locomotor networks in the salamander

Andrej Bicanski, Auke Ijspeert

The evolutionary transition from water to land required new locomotor modes and corresponding adjustments of the spinal "central pattern generators" for locomotion. Salamanders resemble the first terrestrial tetrapods and represent a key animal for the study of these changes. Based on recent physiological data from salamanders, and previous work on the swimming, limbless lamprey, we present a model of the basic oscillatory network in the salamander spinal cord, the spinal segment. Model neurons are of the Hodgkin-Huxley type. Spinal hemisegments contain sparsely connected excitatory and inhibitory neuron populations, and are coupled to a contralateral hemisegment. The model yields a large range of experimental findings, especially the NMDA-induced oscillations observed in isolated axial hemisegments and segments of the salamander Pleurodeles waltlii. The model reproduces most of the effects of the blockade of AMPA synapses, glycinergic synapses, calcium-activated potassium current, persistent sodium current, and -current. Driving segments with a population of brainstem neurons yields fast oscillations in the in vivo swimming frequency range. A minimal modification to the conductances involved in burst-termination yields the slower stepping frequency range. Slow oscillators can impose their frequency on fast oscillators, as is likely the case during gait transitions from swimming to stepping. Our study shows that a lamprey-like network can potentially serve as a building block of axial and limb oscillators for swimming and stepping in salamanders.

Springer Berlin Heidelberg2013

How single neuron properties shape chaotic dynamics and signal transmission in random neural networks

Wulfram Gerstner, Samuel Pavio Muscinelli, Tilo Schwalger

While most models of randomly connected neural networks assume single-neuron models with simple dynamics, neurons in the brain exhibit complex intrinsic dynamics over multiple timescales. We analyze how the dynamical properties of single neurons and recurrent connections interact to shape the effective dynamics in large randomly connected networks. A novel dynamical mean-field theory for strongly connected networks of multi-dimensional rate neurons shows that the power spectrum of the network activity in the chaotic phase emerges from a nonlinear sharpening of the frequency response function of single neurons. For the case of two-dimensional rate neurons with strong adaptation, we find that the network exhibits a state of resonant chaos, characterized by robust, narrow-band stochastic oscillations. The coherence of stochastic oscillations is maximal at the onset of chaos and their correlation time scales with the adaptation timescale of single units. Surprisingly, the resonance frequency can be predicted from the properties of isolated neurons, even in the presence of heterogeneity in the adaptation parameters. In the presence of these internally-generated chaotic fluctuations, the transmission of weak, low-frequency signals is strongly enhanced by adaptation, whereas signal transmission is not influenced by adaptation in the non-chaotic regime. Our theoretical framework can be applied to other mechanisms at the level of single neurons, such as synaptic filtering, refractoriness or spike synchronization. These results advance our understanding of the interaction between the dynamics of single units and recurrent connectivity, which is a fundamental step toward the description of biologically realistic neural networks. Author summary Biological neural networks are formed by a large number of neurons whose interactions can be extremely complex. Such systems have been successfully studied using random network models, in which the interactions among neurons are assumed to be random. However, the dynamics of single units are usually described using over-simplified models, which might not capture several salient features of real neurons. Here, we show how accounting for richer single-neuron dynamics results in shaping the network dynamics and determines which signals are better transmitted. We focus on adaptation, an important mechanism present in biological neurons that consists in the decrease of their firing rate in response to a sustained stimulus. Our mean-field approach reveals that the presence of adaptation shifts the network into a previously unreported dynamical regime, that we term resonant chaos, in which chaotic activity has a strong oscillatory component. Moreover, we show that this regime is advantageous for the transmission of low-frequency signals. Our work bridges the microscopic dynamics (single neurons) to the macroscopic dynamics (network), and shows how the global signal-transmission properties of the network can be controlled by acting on the single-neuron dynamics. These results paves the way for further developments that include more complex neural mechanisms, and considerably advance our understanding of realistic neural networks.

2019

Estimation of small-world topology of cortical networks using Generalized Linear Models

Joao Emanuel Felipe Gerhard, Wulfram Gerstner

Since the seminal work of Watts & Strogatz and others in the late 90s [1], graph-theoretic analyses have been performed on many complex dynamic networks, including brain structures. Most studies have focused on functional connectivity defined between whole brain regions, using imaging techniques such as fMRI, EEG or MEG [2]. Only very few studies have attempted to look at the structure of neural networks on the level of individual neurons [3,4]. To the best of our knowledge, these studies have only considered undirected connectivity networks and have derived connectivity based on estimates on small subsets or even pairs of neurons from the recorded networks. Here, we investigate scale-free and small-world properties of neuronal networks, based on multi-electrode recordings from the awake monkey on a larger data set than in previous approaches. We estimate effective, i.e. causal, interactions by fitting Generalized Linear Models on the neural responses to natural stimulation, incorporating effects from the neurons’ self-history, coupling terms and modulation by the external stimulus. The resulting connectivity matrix is directed and a link between two neurons represents a causal influence from one neuron to the other, given the observation of all other neurons from the population. We use this connectivity matrix to estimate scale-free and small-world properties of the network samples. For this, the quantity proposed by [5] for quantifying small-world-ness is generalized to directed networks. We find that the networks under consideration lack scale-free behavior, but show a small, but significant small-world structure. Finally, we show that the experimental design of multi-electrode recordings typically enforces a particular network structure that can have a considerate impact on how the small-world structure of the network should be evaluated. Since for multi-electrode recordings the sampling of neurons is not uniform across the population (one electrode usually captures signals from a small local population), we expect the wiring probability between neurons of the same local spot to be much higher than the wiring probability between neurons from different electrodes. Thus, random graphs that take the different wiring probabilities into account can serve as a more refined null model than the homogeneous Erdös-Renyi random graphs that are usually proposed as reference models to evaluate small-world properties. Moreover, as the set of recorded neurons in a given experiment always represents only a subpopulation of the overall network, we investigate how the evaluation of small-world structure is affected by this undersampling bias.