Simulation study on effects of channel noise on differential conduction at an axon branch

Branching morphology determines signal propagation dynamics in neurons

Computational modeling of signal propagation in neurons is critical to our understanding of basic principles underlying brain organization and activity. Exploring these models is used to address basic neuroscience questions as well as to gain insights for clinical applications. The seminal Hodgkin Huxley model is a common theoretical framework to study brain activity. It was mainly used to investigate the electrochemical and physical properties of neurons. The influence of neuronal structure on activity patterns was explored, however, the rich dynamics observed in neurons with different morphologies is not yet fully understood. Here, we study signal propagation in fundamental building blocks of neuronal branching trees, unbranched and branched axons. We show how these simple axonal elements can code information on spike trains, and how asymmetric responses can emerge in axonal branching points. This asymmetric phenomenon has been observed experimentally but until now lacked theoretical characterization. Together, our results suggest that axonal morphological parameters are instrumental in activity modulation and information coding. The insights gained from this work lay the ground for better understanding the interplay between function and form in real-world complex systems. It may also supply theoretical basis for the development of novel therapeutic approaches to damaged nervous systems.

Spike propagation in dendrites with stochastic ion channels

We investigate the effects of the stochastic nature of ion channels on the faithfulness, precision and reproducibility of electrical signal transmission in weakly active, dendritic membrane under in vitro conditions. The properties of forward and backpropagating action potentials (BPAPs) in the dendritic tree of pyramidal cells are the subject of intense empirical work and theoretical speculation (Larkum et al., 1999; Zhu, 2000; Larkum et al., 2001; Larkum and Zhu, 2002; Schaefer et al., 2003; Williams, 2004; Waters et al., 2005). We numerically simulate the effects of stochastic ion channels on the forward and backward propagation of dendritic spikes in Monte-Carlo simulations on a reconstructed layer 5 pyramidal neuron. We report that in most instances there is little variation in timing or amplitude for a single BPAP, while variable backpropagation can occur for trains of action potentials. Additionally, we find that the generation and forward propagation of dendritic Ca(2+) spikes are susceptible to channel variability. This indicates limitations on computations that depend on the precise timing of Ca(2+) spikes.

Effects of patch temperature on spontaneous action potential train due to channel fluctuations: Coherence resonance

Based on the Hodgkin-Huxley (HH) model, the effects of patch temperature as a control parameter on the spontaneous action potentials for finite size of membrane patch are studied. With increasing patch temperature, it is found that the mean open rates of sodium and potassium channels of the HH neuron are decreased, and the mean duration of spikes of membrane potential is also decreased, which are qualitatively consistent with previous experimental results of single ion channel. Under moderate patch size, the mean interspike interval of membrane potential first decreases, reaches a minimum, and then increases with increasing patch temperature. It is shown that for both low and high temperatures, the channels fluctuation-induced spontaneous action potentials appear to be rather irregular, while for moderate patch temperature, relatively coherent oscillations observed. By defining a measure parameter beta, we show that there is a maximal region for the measure beta in the patch temperature and patch size parameter plane where the coherence resonance phenomena are very remarkable, and the characteristic correlation time of the output also confirm our result.

Rallian "equivalent" cylinders reconsidered: comparisons with literal compartments

In Rall's "equivalent" cylinder morphological-to-electrical transformation, neuronal arborizations are reduced to single unbranched core-conductors. The conventional assumption that such an "equivalent" reconstructs the electrical properties of the fibers it represents was tested directly; electrical properties and responses of "equivalent" cylinders were compared with those of their literal branch constituents for fibers with a single symmetrical bifurcation. The numerical solution methods were validated independently by their accurate reconstruction of the responses of an analog circuit configured with compartmental architecture to solve the cable equation for passive fibers with a symmetrical bifurcation. In passive fibers, "equivalent" cylinders misestimated the spatial distribution of voltage amplitudes and steady-state input resistance, partly due to the lack of axial current bifurcation. In active fibers with a single propagating action potential, the spatial distributions of point-to-point conduction velocity values (measured in meters/second) for a literal branch point differed significantly from those of their "equivalent" cylinders. "Equivalent" cylinders also underestimated the diameter-dependent delay in propagation through the branch point and branches, due to the larger "equivalent" diameter. Corrections to the "equivalent" cylinder did not reconcile differences between "equivalent" and literal models. However, "equivalent" and literal branch fibers had the same (a) steady-state resistance "looking into" an isolated symmetrical branch point and (b) geometry-independent point-to-point propagation velocity when measured in space constants per millisecond except within +/-1 space constant from the geometrical inhomogeneity. In summary, Rall's "equivalent" cylinders did not accurately reconstruct all passive or active electrophysiological properties and responses of their literal compartments. For the modeling of individual neurons, the requirement of single-branch resolution is discussed.

Coherence resonance with multiple peaks in a coupled FitzHugh-Nagumo model

Coherence resonance in a coupled excitable system is studied, through both a computer simulation and a circuit experiment, using a piecewise linear version of the FitzHugh-Nagumo model. White noise is added to the first element and spikes are accordingly generated and transmitted to the second element. The mean, standard deviation, and coefficient of variation of the interspike intervals in the second element have multiple peaks as the noise strengthens, when the coupling strength is small. This results from the phase locking to the spikes that the noise generates.

Computer model for action potential propagation through branch point in myelinated nerves

A mathematical model is developed for simulation of action potential propagation through a single branch point of a myelinated nerve fiber with a parent branch bifurcating into two identical daughter branches. This model is based on a previously published multi-layer compartmental model for single unbranched myelinated nerve fibers. Essential modifications were made to couple both daughter branches to the parent branch. There are two major features in this model. First, the model could incorporate detailed geometrical parameters for the myelin sheath and the axon, accomplished by dividing both structures into many segments. Second, each segment has two layers, the myelin sheath and the axonal membrane, allowing voltages of intra-axonal space and periaxonal space to be calculated separately. In this model, K ion concentration in the periaxonal space is dynamically linked to the activity of axonal fast K channels underneath the myelin in the paranodal region. Our model demonstrates that the branch point acts like a low-pass filter, blocking high-frequency transmission from the parent to the daughter branches. Theoretical analysis showed that the cutoff frequency for transmission through the branch point is determined by temperature, local K ion accumulation, width of the periaxonal space, and internodal lengths at the vicinity of the branch point. Our result is consistent with empirical findings of irregular spacing of nodes of Ranvier at axon abors, suggesting that branch points of myelinated axons play important roles in signal integration in an axonal tree.

Subthreshold voltage noise due to channel fluctuations in active neuronal membranes

Voltage-gated ion channels in neuronal membranes fluctuate randomly between different conformational states due to thermal agitation. Fluctuations between conducting and nonconducting states give rise to noisy membrane currents and subthreshold voltage fluctuations and may contribute to variability in spike timing. Here we study subthreshold voltage fluctuations due to active voltage-gated Na+ and K+ channels as predicted by two commonly used kinetic schemes: the Mainen et al. (1995) (MJHS) kinetic scheme, which has been used to model dendritic channels in cortical neurons, and the classical Hodgkin-Huxley (1952) (HH) kinetic scheme for the squid giant axon. We compute the magnitudes, amplitude distributions, and power spectral densities of the voltage noise in isopotential membrane patches predicted by these kinetic schemes. For both schemes, noise magnitudes increase rapidly with depolarization from rest. Noise is larger for smaller patch areas but is smaller for increased model temperatures. We contrast the results from Monte Carlo simulations of the stochastic nonlinear kinetic schemes with analytical, closed-form expressions derived using passive and quasi-active linear approximations to the kinetic schemes. For all subthreshold voltage ranges, the quasi-active linearized approximation is accurate within 8% and may thus be used in large-scale simulations of realistic neuronal geometries.

Axons as computing devices: basic insights gained from models

Detailed models of single neurons are typically focused on the dendritic tree and ignore the axonal tree, assuming that the axon is a simple transmission line. In the last 40 years, however, several theoretical and experimental studies have suggested that axons could implement information processing tasks by exploiting: 1) the time delay in action potential (AP) propagation along the axon; 2) the differential filtering of APs into the axonal subtrees; and 3) their activity-dependent excitability. Models for axonal trees have attempted to examine the feasibility of these ideas. However, because the physiological and anatomical data on axons are seriously limited, realistic models of axons have not been developed. The present paper summarizes the main insights that were gained from simplified models of axons; it also highlights the stochastic nature of axons, a topic that was largely neglected in classical models of axons. The advance of new experimental techniques makes it now possible to pay a very close experimental visit to axons. Theoretical tools and fast computers enable to go beyond the simplified models and to construct realistic models of axons. When tightly linked, experiments and theory will help to unravel how axons share the information processing tasks that single neurons implement.

Ion channel stochasticity may be critical in determining the reliability and precision of spike timing

The firing reliability and precision of an isopotential membrane patch consisting of a realistically large number of ion channels is investigated using a stochastic Hodgkin-Huxley (HH) model. In sharp contrast to the deterministic HH model, the biophysically inspired stochastic model reproduces qualitatively the different reliability and precision characteristics of spike firing in response to DC and fluctuating current input in neocortical neurons, as reported by Mainen & Sejnowski (1995). For DC inputs, spike timing is highly unreliable; the reliability and precision are significantly increased for fluctuating current input. This behavior is critically determined by the relatively small number of excitable channels that are opened near threshold for spike firing rather than by the total number of channels that exist in the membrane patch. Channel fluctuations, together with the inherent bistability in the HH equations, give rise to three additional experimentally observed phenomena: subthreshold oscillations in the membrane voltage for DC input, "spontaneous" spikes for subthreshold inputs, and "missing" spikes for suprathreshold inputs. We suggest that the noise inherent in the operation of ion channels enables neurons to act as "smart" encoders. Slowly varying, uncorrelated inputs are coded with low reliability and accuracy and, hence, the information about such inputs is encoded almost exclusively by the spike rate. On the other hand, correlated presynaptic activity produces sharp fluctuations in the input to the postsynaptic cell, which are then encoded with high reliability and accuracy. In this case, information about the input exists in the exact timing of the spikes. We conclude that channel stochasticity should be considered in realistic models of neurons.