We study the dynamics of a noisy network of spiking neurons with spike-frequency adaptation (SFA), using a mean-field approach, in terms of a two-dimensional Fokker-Planck equation for the membrane potential of the neurons and the calcium concentration gating SFA. The long time scales of SFA allow us to use an adiabatic approximation and to describe the network as an effective nonlinear two-dimensional system. The phase diagram is computed for varying levels of SFA and synaptic coupling. Two different population-bursting regimes emerge, depending on the level of SFA in networks with noisy emission rate, due to the finite number of neurons.
We propose a novel explanation for bistable perception, namely, the collective dynamics of multiple neural populations that are individually meta-stable. Distributed representations of sensory input and of perceptual state build gradually through noise-driven transitions in these populations, until the competition between alternative representations is resolved by a threshold mechanism. The perpetual repetition of this collective race to threshold renders perception bistable. This collective dynamics – which is largely uncoupled from the time-scales that govern individual populations or neurons – explains many hitherto puzzling observations about bistable perception: the wide range of mean alternation rates exhibited by bistable phenomena, the consistent variability of successive dominance periods, and the stabilizing effect of past perceptual states. It also predicts a number of previously unsuspected relationships between observable quantities characterizing bistable perception. We conclude that bistable perception reflects the collective nature of neural decision making rather than properties of individual populations or neurons.
Cortical networks, in-vitro as well as in-vivo, can spontaneously generate a variety of collective dynamical events such as network spikes, UP and DOWN states, global oscillations, and avalanches. Though each of them has been variously recognized in previous works as expression of the excitability of the cortical tissue and the associated nonlinear dynamics, a unified picture of the determinant factors (dynamical and architectural) is desirable and not yet available. Progress has also been partially hindered by the use of a variety of statistical measures to define the network events of interest. We propose here a common probabilistic definition of network events that, applied to the firing activity of cultured neural networks, highlights the co-occurrence of network spikes, power-law distributed avalanches, and exponentially distributed ‘quasi-orbits’, which offer a third type of collective behavior. A rate model, including synaptic excitation and inhibition with no imposed topology, synaptic short-term depression, and finite-size noise, accounts for all these different, coexisting phenomena. We find that their emergence is largely regulated by the proximity to an oscillatory instability of the dynamics, where the non-linear excitable behavior leads to a self-amplification of activity fluctuations over a wide range of scales in space and time. In this sense, the cultured network dynamics is compatible with an excitation-inhibition balance corresponding to a slightly sub-critical regime. Finally, we propose and test a method to infer the characteristic time of the fatigue process, from the observed time course of the network’s firing rate. Unlike the model, possessing a single fatigue mechanism, the cultured network appears to show multiple time scales, signalling the possible coexistence of different fatigue mechanisms.
The spike activity of cells in some cortical areas has been found to be correlated with reaction times and behavioral responses during two-choice decision tasks. These experimental findings have motivated the study of biologically plausible winner-take-all network models, in which strong recurrent excitation and feedback inhibition allow the network to form a categorical choice upon stimulation. Choice formation corresponds in these models to the transition from the spontaneous state of the network to a state where neurons selective for one of the choices fire at a high rate and inhibit the activity of the other neurons. This transition has been traditionally induced by an increase in the external input that destabilizes the spontaneous state of the network and forces its relaxation to a decision state. Here we explore a different mechanism by which the system can undergo such transitions while keeping the spontaneous state stable, based on an escape induced by finite-size noise from the spontaneous state. This decision mechanism naturally arises for low stimulus strengths and leads to exponentially distributed decision times when the amount of noise in the system is small. Furthermore, we show using numerical simulations that mean decision times follow in this regime an exponential dependence on the amplitude of noise. The escape mechanism provides thus a dynamical basis for the wide range and variability of decision times observed experimentally.
Echo state networks (ESNs) are a powerful form of reservoir computing that only require training of linear output weights while the internal reservoir is formed of fixed randomly connected neurons. With a correctly scaled connectivity matrix, the neurons’ activity exhibits the echo-state property and responds to the input dynamics with certain timescales. Tuning the timescales of the network can be necessary for treating certain tasks, and some environments require multiple timescales for an efficient representation. Here we explore the timescales in hierarchical ESNs, where the reservoir is partitioned into two smaller linked reservoirs with distinct properties. Over three different tasks (NARMA10, a reconstruction task in a volatile environment, and psMNIST), we show that by selecting the hyper-parameters of each partition such that they focus on different timescales, we achieve a significant performance improvement over a single ESN. Through a linear analysis, and under the assumption that the timescales of the first partition are much shorter than the second’s (typically corresponding to optimal operating conditions), we interpret the feedforward coupling of the partitions in terms of an effective representation of the input signal, provided by the first partition to the second, whereby the instantaneous input signal is expanded into a weighted combination of its time derivatives. Furthermore, we propose a data-driven approach to optimise the hyper-parameters through a gradient descent optimisation method that is an online approximation of backpropagation through time. We demonstrate the application of the online learning rule across all the tasks considered.
Biological networks display a variety of activity patterns reflecting a web of interactions that is complex both in space and time. Yet inference methods have mainly focused on reconstructing, from the network’s activity, the spatial structure, by assuming equilibrium conditions or, more recently, a probabilistic dynamics with a single arbitrary time-step. Here we show that, under this latter assumption, the inference procedure fails to reconstruct the synaptic matrix of a network of integrate-and-fire neurons when the chosen time scale of interaction does not closely match the synaptic delay or when no single time scale for the interaction can be identified; such failure, moreover, exposes a distinctive bias of the inference method that can lead to infer as inhibitory the excitatory synapses with interaction time scales longer than the model’s time-step. We therefore introduce a new two-step method, that first infers through cross-correlation profiles the delay-structure of the network and then reconstructs the synaptic matrix, and successfully test it on networks with different topologies and in different activity regimes. Although step one is able to accurately recover the delay-structure of the network, thus getting rid of any a priori guess about the time scales of the interaction, the inference method introduces nonetheless an arbitrary time scale, the time-bin dt used to binarize the spike trains. We therefore analytically and numerically study how the choice of dt affects the inference in our network model, finding that the relationship between the inferred couplings and the real synaptic efficacies, albeit being quadratic in both cases, depends critically on dt for the excitatory synapses only, whilst being basically independent of it for the inhibitory ones.
Inference methods are widely used to recover effective models from observed data. However, few studies attempted to investigate the dynamics of inferred models in neuroscience, and none, to our knowledge, at the network level. We introduce a principled modification of a widely used generalized linear model (GLM), and learn its structural and dynamic parameters from in-vitro spike data. The spontaneous activity of the new model captures prominent features of the non-stationary and non-linear dynamics displayed by the biological network, where the reference GLM largely fails, and also reflects fine-grained spatio-temporal dynamical features. Two ingredients were key for success. The first is a saturating transfer function: beyond its biological plausibility, it limits the neuron’s information transfer, improving robustness against endogenous and external noise. The second is a super-Poisson spikes generative mechanism; it accounts for the undersampling of the network, and allows the model neuron to flexibly incorporate the observed activity fluctuations.
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