Theoretical connections between mathematical neuronal models corresponding to different expressions of noise

A framework for macroscopic phase-resetting curves for generalised spiking neural networks

Brain rhythms emerge from synchronization among interconnected spiking neurons. Key properties of such rhythms can be gleaned from the phase-resetting curve (PRC). Inferring the PRC and developing a systematic phase reduction theory for large-scale brain rhythms remains an outstanding challenge. Here we present a theoretical framework and methodology to compute the PRC of generic spiking networks with emergent collective oscillations. We adopt a renewal approach where neurons are described by the time since their last action potential, a description that can reproduce the dynamical feature of many cell types. For a sufficiently large number of neurons, the network dynamics are well captured by a continuity equation known as the refractory density equation. We develop an adjoint method for this equation giving a semi-analytical expression of the infinitesimal PRC. We confirm the validity of our framework for specific examples of neural networks. Our theoretical framework can link key biological properties at the individual neuron scale and the macroscopic oscillatory network properties. Beyond spiking networks, the approach is applicable to a broad class of systems that can be described by renewal processes.

The formation of oscillatory neuronal assemblies at the network level has been hypothesized to be fundamental to many cognitive and motor functions. One prominent tool to understand the dynamics of oscillatory activity response to stimuli, and hence the neural code for which it is a substrate, is a nonlinear measure called Phase-Resetting Curve (PRC). At the network scale, the PRC defines the measure of how a given synaptic input perturbs the timing of next upcoming volley of spike assemblies: either advancing or delaying this timing. As a further application, one can use PRCs to make unambiguous predictions about whether communicating networks of neurons will phase-lock as it is often observed across the cortical areas and what would be this stable phase-configuration: synchronous, asynchronous or with asymmetric phase-shifts. The latter configuration also implies a preferential flow of information form the leading network to the follower, thereby giving causal signatures of directed functional connectivity. Because of the key position of the PRC in studying synchrony, information flow and entrainment to external forcing, it is crucial to move toward a theory that allows to compute the PRCs of network-wide oscillations not only for a restricted class of models, as has been done in the past, but to network descriptions that are generalized and can reflect flexibly single cell properties. In this manuscript, we tackle this issue by showing how the PRC for network oscillations can be computed using the adjoint systems of partial differential equations that define the dynamics of the neural activity density.

Mapping input noise to escape noise in integrate-and-fire neurons: a level-crossing approach

Noise in spiking neurons is commonly modeled by a noisy input current or by generating output spikes stochastically with a voltage-dependent hazard rate ("escape noise"). While input noise lends itself to modeling biophysical noise processes, the phenomenological escape noise is mathematically more tractable. Using the level-crossing theory for differentiable Gaussian processes, we derive an approximate mapping between colored input noise and escape noise in leaky integrate-and-fire neurons. This mapping requires the first-passage-time (FPT) density of an overdamped Brownian particle driven by colored noise with respect to an arbitrarily moving boundary. Starting from the Wiener-Rice series for the FPT density, we apply the second-order decoupling approximation of Stratonovich to the case of moving boundaries and derive a simplified hazard-rate representation that is local in time and numerically efficient. This simplification requires the calculation of the non-stationary auto-correlation function of the level-crossing process: For exponentially correlated input noise (Ornstein-Uhlenbeck process), we obtain an exact formula for the zero-lag auto-correlation as a function of noise parameters, mean membrane potential and its speed, as well as an exponential approximation of the full auto-correlation function. The theory well predicts the FPT and interspike interval densities as well as the population activities obtained from simulations with colored input noise and time-dependent stimulus or boundary. The agreement with simulations is strongly enhanced across the sub- and suprathreshold firing regime compared to a first-order decoupling approximation that neglects correlations between level crossings. The second-order approximation also improves upon a previously proposed theory in the subthreshold regime. Depending on a simplicity-accuracy trade-off, all considered approximations represent useful mappings from colored input noise to escape noise, enabling progress in the theory of neuronal population dynamics.

Order Through Disorder: The Characteristic Variability of Systems

Randomness characterizes many processes in nature, and therefore its importance cannot be overstated. In the present study, we investigate examples of randomness found in various fields, to underlie its fundamental processes. The fields we address include physics, chemistry, biology (biological systems from genes to whole organs), medicine, and environmental science. Through the chosen examples, we explore the seemingly paradoxical nature of life and demonstrate that randomness is preferred under specific conditions. Furthermore, under certain conditions, promoting or making use of variability-associated parameters may be necessary for improving the function of processes and systems.

Mind the last spike - firing rate models for mesoscopic populations of spiking neurons

The dominant modeling framework for understanding cortical computations are heuristic firing rate models. Despite their success, these models fall short to capture spike synchronization effects, to link to biophysical parameters and to describe finite-size fluctuations. In this opinion article, we propose that the refractory density method (RDM), also known as age-structured population dynamics or quasi-renewal theory, yields a powerful theoretical framework to build rate-based models for mesoscopic neural populations from realistic neuron dynamics at the microscopic level. We review recent advances achieved by the RDM to obtain efficient population density equations for networks of generalized integrate-and-fire (GIF) neurons - a class of neuron models that has been successfully fitted to various cell types. The theory not only predicts the nonstationary dynamics of large populations of neurons but also permits an extension to finite-size populations and a systematic reduction to low-dimensional rate dynamics. The new types of rate models will allow a re-examination of models of cortical computations under biological constraints.