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Person# Joao Emanuel Felipe Gerhard

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Neural oscillations, or brainwaves, are rhythmic or repetitive patterns of neural activity in the central nervous system. Neural tissue can generate oscillatory activity in many ways, driven either

A neural network can refer to a neural circuit of biological neurons (sometimes also called a biological neural network), a network of artificial neurons or nodes in the case of an artificial neur

Biological neuron models, also known as a spiking neuron models, are mathematical descriptions of the properties of certain cells in the nervous system that generate sharp electrical potentials acr

To appreciate how neural circuits in the brain control behaviors, we must identify how the neurons comprising the circuit are connected. Neuronal connectivity is difficult to determine experimentally, whereas neuronal activity can often be readily measured. I describe a statistical framework to estimate circuit connectivity directly from measured activity patterns. Because we usually only have access to a small subset of neurons of a circuit, the estimated connectivity reflects an effective coupling, that is, how spiking activity in one neuron effectively modulates the activity of other neurons. For small circuits, like the nervous system of the crab that controls gut muscle activity, we could show that it is possible to derive the actual physiological connectivity from observing neural activity alone. This was achieved with a regression model adapted to the spike train structure of the data (Generalized Linear Model, GLM). This is the first successful demonstration of a network inference algorithm on a physiological circuit for which the connections are known. For larger networks, like cortical networks, the concept of effective connectivity - though not equivalent to structural connectivity - is useful to characterize the functional properties of the network. For example, we may assess whether networks have small-world or scale-free properties that are important for information processing. We find that cortical networks show a small, but significant small-world structure by applying our estimation framework on multi-electrode recordings from the visual system of the awake monkey. Finally, we study how well spike dynamics and network topology can be inferred from noisy calcium imaging data. We applied our framework on simulated data to explore how uncertainties in spike inference due to experimental parameters affect estimates of network connectivity and their topological features. We find that considerable information about the connectivity can be extracted from the neural activity, but only if spikes are reconstructed with high temporal precision. We then study how errors in the network reconstruction affect the estimation of a number of graph-theoretic measures. Our findings provide a benchmark for future experiments that aim to reliably infer neuronal network properties.

Moritz Deger, Joao Emanuel Felipe Gerhard

Point process generalized linear models (PP-GLMs) provide an important statistical framework for modeling spiking activity in single-neurons and neuronal networks. Stochastic stability is essential when sampling from these models, as done in computational neuroscience to analyze statistical properties of neuronal dynamics and in neuro-engineering to implement closed-loop applications. Here we show, however, that despite passing common goodness-of-fit tests, PP-GLMs estimated from data are often unstable, leading to divergent firing rates. The inclusion of absolute refractory periods is not a satisfactory solution since the activity then typically settles into unphysiological rates. To address these issues, we derive a framework for determining the existence and stability of fixed points of the expected conditional intensity function (CIF) for general PP-GLMs. Specifically, in nonlinear Hawkes PP-GLMs, the CIF is expressed as a function of the previous spike history and exogenous inputs. We use a mean-field quasi-renewal (QR) approximation that decomposes spike history effects into the contribution of the last spike and an average of the CIF over all spike histories prior to the last spike. Fixed points for stationary rates are derived as self-consistent solutions of integral equations. Bifurcation analysis and the number of fixed points predict that the original models can show stable, divergent, and metastable (fragile) dynamics. For fragile models, fluctuations of the single-neuron dynamics predict expected divergence times after which rates approach unphysiologically high values. This metric can be used to estimate the probability of rates to remain physiological for given time periods, e.g., for simulation purposes. We demonstrate the use of the stability framework using simulated single-neuron examples and neurophysiological recordings. Finally, we show how to adapt PPGLM estimation procedures to guarantee model stability. Overall, our results provide a stability framework for data-driven PP-GLMs and shed new light on the stochastic dynamics of state-of-the-art statistical models of neuronal spiking activity.

Joao Emanuel Felipe Gerhard, Wulfram Gerstner, Friedemann Zenke

Two-photon calcium imaging enables functional analysis of neuronal circuits by inferring action potential (AP) occurrence ("spike trains") from cellular fluorescence signals. It remains unclear how experimental parameters such as signal-to-noise ratio (SNR) and acquisition rate affect spike inference and whether additional information about network structure can be extracted. Here we present a simulation framework for quantitatively assessing how well spike dynamics and network topology can be inferred from noisy calcium imaging data. For simulated AP-evoked calcium transients in neocortical pyramidal cells, we analyzed the quality of spike inference as a function of SNR and data acquisition rate using a recently introduced peeling algorithm. Given experimentally attainable values of SNR and acquisition rate, neural spike trains could be reconstructed accurately and with up to millisecond precision. We then applied statistical neuronal network models to explore how remaining uncertainties in spike inference affect estimates of network connectivity and topological features of network organization. We define the experimental conditions suitable for inferring whether the network has a scale-free structure and determine how well hub neurons can be identified. Our findings provide a benchmark for future calcium imaging studies that aim to reliably infer neuronal network properties.