Effects of passive dendritic tree properties on the firing dynamics of a leaky-integrate-and-fire neuron

Spike-Conducting Integrate-and-Fire Model

Modeling is a useful tool for investigating various biophysical characteristics of neurons. Recent simulation studies of propagating action potentials (spike conduction) along axons include the investigation of neuronal activity evoked by electrical stimulation from implantable prosthetic devices. In contrast to point-neuron simulations, where a large variety of models are readily available, Hodgkin–Huxley-type conductance-based models have been almost the only option for simulating axonal spike conduction, as simpler models cannot faithfully replicate the waveforms of propagating spikes. Since the amount of available physiological data, especially in humans, is usually limited, calibration, and justification of the large number of parameters of a complex model is generally difficult. In addition, not all simulation studies of axons require detailed descriptions of nonlinear ionic dynamics. In this study, we construct a simple model of spike generation and conduction based on the exponential integrate-and-fire model, which can simulate the rapid growth of the membrane potential at spike initiation. In terms of the number of parameters and equations, this model is much more compact than conventional models, but can still reliably simulate spike conduction along myelinated and unmyelinated axons that are stimulated intracellularly or extracellularly. Our simulations of auditory nerve fibers with this new model suggest that, because of the difference in intrinsic membrane properties, the axonal spike conduction of high-frequency nerve fibers is faster than that of low-frequency fibers. The simple model developed in this study can serve as a computationally efficient alternative to more complex models for future studies, including simulations of neuroprosthetic devices.

Heterogeneous firing responses predict diverse couplings to presynaptic activity in mice layer V pyramidal neurons

In this study, we present a theoretical framework combining experimental characterizations and analytical calculus to capture the firing rate input-output properties of single neurons in the fluctuation-driven regime. Our framework consists of a two-step procedure to treat independently how the dendritic input translates into somatic fluctuation variables, and how the latter determine action potential firing. We use this framework to investigate the functional impact of the heterogeneity in firing responses found experimentally in young mice layer V pyramidal cells. We first design and calibrate in vitro a simplified morphological model of layer V pyramidal neurons with a dendritic tree following Rall's branching rule. Then, we propose an analytical derivation for the membrane potential fluctuations at the soma as a function of the properties of the synaptic input in dendrites. This mathematical description allows us to easily emulate various forms of synaptic input: either balanced, unbalanced, synchronized, purely proximal or purely distal synaptic activity. We find that those different forms of dendritic input activity lead to various impact on the somatic membrane potential fluctuations properties, thus raising the possibility that individual neurons will differentially couple to specific forms of activity as a result of their different firing response. We indeed found such a heterogeneous coupling between synaptic input and firing response for all types of presynaptic activity. This heterogeneity can be explained by different levels of cellular excitability in the case of the balanced, unbalanced, synchronized and purely distal activity. A notable exception appears for proximal dendritic inputs: increasing the input level can either promote firing response in some cells, or suppress it in some other cells whatever their individual excitability. This behavior can be explained by different sensitivities to the speed of the fluctuations, which was previously associated to different levels of sodium channel inactivation and density. Because local network connectivity rather targets proximal dendrites, our results suggest that this aspect of biophysical heterogeneity might be relevant to neocortical processing by controlling how individual neurons couple to local network activity.

Multiscale modeling of brain dynamics: from single neurons and networks to mathematical tools

The extreme complexity of the brain naturally requires mathematical modeling approaches on a large variety of scales; the spectrum ranges from single neuron dynamics over the behavior of groups of neurons to neuronal network activity. Thus, the connection between the microscopic scale (single neuron activity) to macroscopic behavior (emergent behavior of the collective dynamics) and vice versa is a key to understand the brain in its complexity. In this work, we attempt a review of a wide range of approaches, ranging from the modeling of single neuron dynamics to machine learning. The models include biophysical as well as data-driven phenomenological models. The discussed models include Hodgkin-Huxley, FitzHugh-Nagumo, coupled oscillators (Kuramoto oscillators, Rössler oscillators, and the Hindmarsh-Rose neuron), Integrate and Fire, networks of neurons, and neural field equations. In addition to the mathematical models, important mathematical methods in multiscale modeling and reconstruction of the causal connectivity are sketched. The methods include linear and nonlinear tools from statistics, data analysis, and time series analysis up to differential equations, dynamical systems, and bifurcation theory, including Granger causal connectivity analysis, phase synchronization connectivity analysis, principal component analysis (PCA), independent component analysis (ICA), and manifold learning algorithms such as ISOMAP, and diffusion maps and equation-free techniques. WIREs Syst Biol Med 2016, 8:438-458. doi: 10.1002/wsbm.1348 For further resources related to this article, please visit the WIREs website.