Theoretical analysis of the amplification of synaptic potentials by small clusters of persistent sodium channels in dendrites

A nonlinear cable framework for bidirectional synaptic plasticity

Finding the rules underlying how axons of cortical neurons form neural circuits and modify their corresponding synaptic strength is the still subject of intense research. Experiments have shown that internal calcium concentration, and both the precise timing and temporal order of pre and postsynaptic action potentials, are important constituents governing whether the strength of a synapse located on the dendrite is increased or decreased. In particular, previous investigations focusing on spike timing-dependent plasticity (STDP) have typically observed an asymmetric temporal window governing changes in synaptic efficacy. Such a temporal window emphasizes that if a presynaptic spike, arriving at the synaptic terminal, precedes the generation of a postsynaptic action potential, then the synapse is potentiated; however if the temporal order is reversed, then depression occurs. Furthermore, recent experimental studies have now demonstrated that the temporal window also depends on the dendritic location of the synapse. Specifically, it was shown that in distal regions of the apical dendrite, the magnitude of potentiation was smaller and the window for depression was broader, when compared to observations from the proximal region of the dendrite. To date, the underlying mechanism(s) for such a distance-dependent effect is (are) currently unknown. Here, using the ionic cable theory framework in conjunction with the standard calcium based plasticity model, we show for the first time that such distance-dependent inhomogeneities in the temporal learning window for STDP can be largely explained by both the spatial and active properties of the dendrite.

Analytical solutions for nonlinear cable equations with calcium dynamics. II. Saltatory transmission in a sparsely excitable cable model

In order to gain a better theoretical understanding of the interaction between voltage and calcium influx, we present the simulation results for saltatory transmission in a sparsely excitable model of a continuous cylindrical segment of nerve fiber, where calcium diffuses internally and various ion channels are distributed as hotspots along the cable. A standard set of ion channel descriptions is used to illustrate how different numbers and distributions of ion channel hotspots affect the propagation and transmission of a single action potential and/or a spike train and how such hotspots affect calcium influx and diffusion within continuous cylindrical segment of nerve fiber.

Influence of electrotonic structure and synaptic mapping on the receptive field properties of a collision-detecting neuron

The lobula giant movement detector (LGMD) is a visual interneuron of Orthopteran insects involved in collision avoidance and escape behavior. The LGMD possesses a large dendritic field thought to receive excitatory, retinotopic projections from the entire compound eye. We investigated whether the LGMD's receptive field for local motion stimuli can be explained by its electrotonic structure and the eye's anisotropic sampling of visual space. Five locust (Schistocerca americana) LGMD neurons were stained and reconstructed. We show that the excitatory dendritic field and eye can be fitted by ellipsoids having similar geometries. A passive compartmental model fit to electrophysiological data was used to demonstrate that the LGMD is not electrotonically compact. We derived a spike rate to membrane potential transform using intracellular recordings under visual stimulation, allowing direct comparison between experimental and simulated receptive field properties. By assuming a retinotopic mapping giving equal weight to each ommatidium and equally spaced synapses, the model reproduced the experimental data along the eye equator, though it failed to reproduce the receptive field along the ventral-dorsal axis. Our results illustrate how interactions between the distribution of synaptic inputs and the electrotonic properties of neurons contribute to shaping their receptive fields.

Analytical solutions for nonlinear cable equations with calcium dynamics. I: Derivations

The interaction between membrane potential and internal calcium concentration plays many important roles in regulating synaptic integration and neuronal firing. In order to gain a better theoretical understanding between the voltage-calcium interaction, a nonlinear cable equation with calcium dynamics is solved analytically. This general reaction-diffusion system represents a model of a cylindrical dendritic segment in which calcium diffuses internally in the presence of buffers, pumps and exchangers, and where ion channels are sparsely distributed over the membrane,in the form of hotspots, acting as point current sources along the dendritic membrane. In order to proceed, the reaction-diffusion system is recast into a system of coupled nonlinear integral equations, with which a perturbative expansion in dimensionless voltage and calcium concentration are used to find analytical solutions to this general system. The resulting solutions can be used to investigate, the interaction between the membrane potential and underlying calcium dynamics in a natural (non-discretized) setting.

fMRI MODELS OF DENDRITIC AND ASTROCYTIC NETWORKS

In order to elucidate the relationships between hierarchical structures within the neocortical neuropil and the information carried by an ensemble of neurons encompassing a single voxel, it is essential to predict through volume conductor modeling LFPs representing average extracellular potentials, which are expressed in terms of interstitial potentials of individual cells in networks of gap-junctionally connected astrocytes and synaptically connected neurons. These relationships have been provided and can then be used to investigate how the underlying neuronal population activity can be inferred from the measurement of the BOLD signal through electrovascular coupling mechanisms across the blood-brain barrier. The importance of both synaptic and extrasynaptic transmission as the basis of electrophysiological indices triggering vascular responses between dendritic and astrocytic networks, and sequential configurations of firing patterns in composite neural networks is emphasized. The purpose of this review is to show how fMRI data may be used to draw conclusions about the information transmitted by individual neurons in populations generating the BOLD signal.

Layer-specific properties of the persistent sodium current in sensorimotor cortex

We evaluated the characteristics of the persistent sodium current (I(NaP)) in pyramidal neurons of layers II/III and V in slices of rat sensorimotor cortex using whole cell patch-clamp recordings. In both layers, I(NaP) began activating around -60 mV and was half-activated at -43 mV. The I(NaP) peak amplitude and density were significantly higher in layer V. The voltage-dependent I(NaP) steady-state inactivation occurred at potentials that were significantly more positive in layer V (V(1/2): -42.3 +/- 1.1 mV) than in layer II/III (V(1/2): -46.8 +/- 1.6 mV). In both layers, a current fraction corresponding to about 25% of the maximal peak amplitude did not inactivate. The time course of I(NaP) inactivation and recovery from inactivation could be fitted with a biexponential function. In layer V pyramidal neurons the faster time constant of development of inactivation had variable values, ranging from 158.0 to 1,133.8 ms, but it was on average significantly slower than that in layer II/III (425.9 +/- 80.5 vs. 145.8 +/- 18.2 ms). In both layers, I(NaP) did not completely inactivate even with very long conditioning depolarizations (40 s at -10 mV). Recovery from inactivation was similar in the two layers. Layer V intrinsically bursting and regular spiking nonadapting neurons showed particularly prolonged depolarized plateau potentials when Ca2+ and K+ currents were blocked and slower early phase of I(NaP) development of inactivation. The biexponential kinetics characterizing the time-dependent inactivation of I(NaP) in layers II/III and V indicates a complex inactivating process that is incomplete, allowing a residual "persistent" current fraction that does not inactivate. Moreover, our data indicate that I(NaP) has uneven inactivation properties in pyramidal neurons of different layers of rat sensorimotor cortex. The higher current density, the rightward shifted voltage dependency of inactivation as well the slower kinetics of inactivation characterizing I(NaP) in layer V with respect to layer II/III pyramidal neurons may play a significant role in their ability to fire recurrent action potential bursts, as well in the high susceptibility to generate epileptic events.

Analytical solutions of the Frankenhaeuser-Huxley equations I: minimal model for backpropagation of action potentials in sparsely excitable dendrites

Hodgkin and Huxley's ionic theory of the nerve impulse embodies principles, applicable also to the impulses in vertebrate nerve fibers, as demonstrated by Bernhard Frankenhaeuser and Andrew Huxley 40 years ago. Frankenhaeuser and Huxley reformulated the classical Hodgkin-Huxley equations, in terms of electrodiffusion theory, and computed action potentials specifically for saltatory conduction in myelinated axons. In this paper, we obtain analytical solutions to the most difficult nonlinear partial differential equations in classical neurophysiology. We solve analytically the Frankenhaeuser-Huxley equations pertaining to a model of sparsely excitable, nonlinear dendrites with clusters of transiently activating, TTX-sensitive Na(+) channels, discretely distributed as point sources of inward current along a continuous (non-segmented) leaky cable structure. Each cluster or hot-spot, corresponding to a mesoscopic level description of Na(+) ion channels, includes known cumulative inactivation kinetics observed at the microscopic level. In such a third-order system, the 'recovery' variable is an electrogenic sodium-pump imbedded in the passive membrane, and the system is stabilized by the presence of a large leak conductance mediated by a composite number of ligand-gated channels permeable to monovalent cations Na(+) and K(+). In order to reproduce antidromic propagation and attenuation of action potentials, a nonlinear integral equation must be solved (in the presence of suprathreshold input, and a constant-field equation of electrodiffusion at each hot-spot with membrane gates controlling the flow of current). A perturbative expansion of the non-dimensional membrane potential (Phi) is used to obtain time-dependent analytical solutions, involving a voltage-dependent Na(+) activation (micro) and a state-dependent inactivation (eta) gating variables. It is shown that action potentials attenuate in amplitude in accordance with experimental findings, and that the spatial density distribution of transient Na(+) channels along a long dendrite contributes significantly to their discharge patterns. A major significance of the analytical modeling, in contrast to the computational modeling of backpropagating action potentials, is the provision of a continuous description of the voltage as a function of position, allowing for greater feasibility in developing large-scale biophysical neural networks, without the need for ad hoc computational modeling.

Dendritic integration in a recurrent network

Classical nonlinear cable theory is appropriate for the unmyelinated axonal membrane because voltage-dependent ion channels are densly distributed, but dendrites with a sparse density distribution of voltage-dependent ion channels show "weakly" excitable membrane properties. Therefore, a model for "weakly" active dendrites is presented by introducing voltage-dependent ion channels at discrete locations along the dendritic cable. This provides an alternative representation for the investigation of regenerative potentials in dendrites in order to explore how active dendrites influence synaptic integration. As an example, we consider a two-neuron recurrent network of biophysically distinct conductance-based model neurons with discrete clusters of persistent sodium channels. Analytical solutions, expressed in terms of a Volterra series expansion for the voltage in response to a suprathreshold input current at the soma of one neuron, are obtained to investigate dendritic spikes, and the effect of backpropagation on distal dendritic spike-like potentials.