Electrophysiological interaction through the interstitial space between adjacent unmyelinated parallel fibers

Signal processing in the vagus nerve: Hypotheses based on new genetic and anatomical evidence

Each organism must regulate its internal state in a metabolically efficient way as it interacts in space and time with an ever-changing and only partly predictable world. Success in this endeavor is largely determined by the ongoing communication between brain and body, and the vagus nerve is a crucial structure in that dialogue. In this review, we introduce the novel hypothesis that the afferent vagus nerve is engaged in signal processing rather than just signal relay. New genetic and structural evidence of vagal afferent fiber anatomy motivates two hypotheses: (1) that sensory signals informing on the physiological state of the body compute both spatial and temporal viscerosensory features as they ascend the vagus nerve, following patterns found in other sensory architectures, such as the visual and olfactory systems; and (2) that ascending and descending signals modulate one another, calling into question the strict segregation of sensory and motor signals, respectively. Finally, we discuss several implications of our two hypotheses for understanding the role of viscerosensory signal processing in predictive energy regulation (i.e., allostasis) as well as the role of metabolic signals in memory and in disorders of prediction (e.g., mood disorders).

Knösche T. Modelling the effect of ephaptic coupling on spike propagation in peripheral nerve fibres

Experimental and theoretical studies have shown that ephaptic coupling leads to the synchronisation and slowing down of spikes propagating along the axons within peripheral nerve bundles. However, the main focus thus far has been on a small number of identical axons, whereas realistic peripheral nerve bundles contain numerous axons with different diameters. Here, we present a computationally efficient spike propagation model, which captures the essential features of propagating spikes and their ephaptic interaction, and facilitates the theoretical investigation of spike volleys in large, heterogeneous fibre bundles. We first lay out the theoretical basis to describe how the spike in an active axon changes the membrane potential of a passive axon. These insights are then incorporated into the spike propagation model, which is calibrated with a biophysically realistic model based on Hodgkin-Huxley dynamics. The fully calibrated model is then applied to fibre bundles with a large number of axons and different types of axon diameter distributions. One key insight of this study is that the heterogeneity of the axonal diameters has a dispersive effect, and that a higher level of heterogeneity requires stronger ephaptic coupling to achieve full synchronisation between spikes.

Modelling neuron fiber interaction and coupling in non-myelinated bundled fiber

Understanding the local dynamics of a neural network relies heavily on local field potential and cell-field interaction. But it is still unclear how local the local potential is and what kinds of consequences the trans-membrane current flow and produced electric field have on the local neural fiber. Mimicking signal transmission in neighboring nerve fiber, a simulation model is built to analyze local behavior due to trans-membrane current, cell-field interactions, and their repercussions on the bundled fiber system. Simulation studies reveal that depending on the coupling parameters, activity in one fiber can depolarize or hyper-polarize adjacent fibers. The suggested cell-field interaction model was tested using an orientation-selective coupled retinal ganglion cell network, which was compared to its uncoupled counterpart. The proposed work has been used to model and simulate local signal dynamics in a bundled fiber system of an orientation-selective RGC network due to cell-field interaction, as well as to gain insight into the possible significance of dendritic fiber coupling in orientation selectivity bandwidth adjustment.

Ephaptic coupling in white matter fibre bundles modulates axonal transmission delays

Axonal connections are widely regarded as faithful transmitters of neuronal signals with fixed delays. The reasoning behind this is that extracellular potentials caused by spikes travelling along axons are too small to have an effect on other axons. Here we devise a computational framework that allows us to study the effect of extracellular potentials generated by spike volleys in axonal fibre bundles on axonal transmission delays. We demonstrate that, although the extracellular potentials generated by single spikes are of the order of microvolts, the collective extracellular potential generated by spike volleys can reach several millivolts. As a consequence, the resulting depolarisation of the axonal membranes increases the velocity of spikes, and therefore reduces axonal delays between brain areas. Driving a neural mass model with such spike volleys, we further demonstrate that only ephaptic coupling can explain the reduction of stimulus latencies with increased stimulus intensities, as observed in many psychological experiments.

Axonal fibre bundles that connect distant cortical areas contain millions of densely packed axons. When synchronous spike volleys travel through such fibre bundles, the extracellular potential within the bundles is perturbed. We use computer simulations to examine the magnitude and shape of this perturbation, and demonstrate that it is sufficiently strong to affect axonal transmission speeds. Since most spikes within a spike volley are positioned in an area where the extracellular potential is negative (relative to a distant reference), the resulting depolarisation of the axonal membranes accelerates the spike volley on average. This finding is in contrast to previous studies of ephaptic coupling effects between axons, where ephaptic coupling was found to slow down spike propagation. Our finding has consequences for information transmission and synchronisation between cortical areas.

A mathematical model of ephaptic interactions in neuronal fiber pathways: Could there be more than transmission along the tracts?

While numerous studies of ephaptic interactions have focused on either axons of peripheral nerves or on cortical structures, no attention has been given to the possibility of ephaptic interactions in white matter tracts. Inspired by the highly organized, tightly packed geometry of axons in fiber pathways, we aim to investigate the potential effects of ephaptic interactions along these structures that are resilient to experimental probing. We use axonal cable theory to derive a minimal model of a sheet of N ephaptically coupled axons. Numerical solutions of the proposed model are explored as ephaptic coupling is varied. We demonstrate that ephaptic interactions can lead to local phase locking between adjacent traveling impulses and that, as coupling is increased, traveling impulses trigger new impulses along adjacent axons, resulting in finite size traveling fronts. For strong enough coupling, impulses propagate laterally and backwards, resulting in complex spatiotemporal patterns. While common large-scale brain network models often model fiber pathways as simple relays of signals between different brain regions, our work calls for a closer reexamination of the validity of such a view. The results suggest that in the presence of significant ephaptic interactions, the brain fiber tracts can act as a dynamic active medium.

Starting from the FitzHugh-Nagumo cable model, we derive a system of nonlinear coupled partial differential equations (PDEs) to model a sheet of N ephaptically coupled axons. We also present a continuous limit approximation transforming the model into a two-dimensional field equation. We numerically solve the equations exploring the dynamics as coupling strength is varied. We observe phase locking of adjacent impulses and coordination of subthreshold dynamics. Strong enough coupling generates complex spatiotemporal patterns as new impulses form traveling fronts propagating laterally and backwards. The transition between different dynamic regimes happens abruptly at critical values of parameter. The results put into question the validity of assuming the role of fiber pathways to be that of mere interneuronal transmission and call for further investigation of the matter.

Modelling the effects of ephaptic coupling on selectivity and response patterns during artificial stimulation of peripheral nerves

Artificial electrical stimulation of peripheral nerves for sensory feedback restoration can greatly benefit from computational models for simulation-based neural implant design in order to reduce the trial-and-error approach usually taken, thus potentially significantly reducing research and development costs and time. To this end, we built a computational model of a peripheral nerve trunk in which the interstitial space between the fibers and the tissues was modelled using a resistor network, thus enabling distance-dependent ephaptic coupling between myelinated axons and between fascicles as well. We used the model to simulate a) the stimulation of a nerve trunk model with a cuff electrode, and b) the propagation of action potentials along the axons. Results were used to investigate the effect of ephaptic interactions on recruitment and selectivity stemming from artificial (i.e., neural implant) stimulation and on the relative timing between action potentials during propagation. Ephaptic coupling was found to increase the number of fibers that are activated by artificial stimulation, thus reducing the artificial currents required for axonal recruitment, and it was found to reduce and shift the range of optimal stimulation amplitudes for maximum inter-fascicular selectivity. During propagation, while fibers of similar diameters tended to lock their action potentials and reduce their conduction velocities, as expected from previous knowledge on bundles of identical axons, the presence of many other fibers of different diameters was found to make their interactions weaker and unstable.

Adaptive spike-artifact removal from local field potentials uncovers prominent beta and gamma band neuronal synchronization

BACKGROUND

Many neurons synchronize their action potentials to the phase of local field potential (LFP) fluctuations in one or more frequency bands. Analyzing this spike-to-LFP synchronization is challenging, however, when neural spikes and LFP are generated in the same local circuit, because the spike's action potential waveform leak into the LFP and distort phase synchrony estimates. Existing approaches to address this spike bleed-through artifact relied on removing the average action potential waveforms of neurons, but this leaves artifacts in the LFP and distorts synchrony estimates.

NEW METHOD

We describe a spike-removal method that surpasses these limitations by decomposing individual action potentials into their frequency components before their removal from the LFP. The adaptively estimated frequency components allow for variable spread, strength and temporal variation of the spike artifact.

RESULTS

This adaptive approach effectively removes spike bleed-through artifacts in synthetic data with known ground truth, and in single neuron and LFP recordings in nonhuman primate striatum. For a large population of neurons with both narrow and broad action potential waveforms, the use of adaptive artifact removal uncovered 20-35 Hz beta and 35-45 Hz gamma band spike-LFP synchronization that would have remained contaminated otherwise.

COMPARISON WITH EXISTING METHODS

We demonstrate that adaptive spike-artifact removal cleans LFP data that remained contaminated when applying existing Bayesian and non-Bayesian methods of average spike-artifact removal.

CONCLUSIONS

Applying adaptive spike-removal from field potentials allows to estimate the phase at which neurons synchronize and the consistency of their phase-locked firing for both beta and low gamma frequencies. These metrics may prove essential to understand cell-to-circuit neuronal interactions in multiple brain systems.

Action potential propagation and synchronisation in myelinated axons

With the advent of advanced MRI techniques it has become possible to study axonal white matter non-invasively and in great detail. Measuring the various parameters of the long-range connections of the brain opens up the possibility to build and refine detailed models of large-scale neuronal activity. One particular challenge is to find a mathematical description of action potential propagation that is sufficiently simple, yet still biologically plausible to model signal transmission across entire axonal fibre bundles. We develop a mathematical framework in which we replace the Hodgkin-Huxley dynamics by a spike-diffuse-spike model with passive sub-threshold dynamics and explicit, threshold-activated ion channel currents. This allows us to study in detail the influence of the various model parameters on the action potential velocity and on the entrainment of action potentials between ephaptically coupled fibres without having to recur to numerical simulations. Specifically, we recover known results regarding the influence of axon diameter, node of Ranvier length and internode length on the velocity of action potentials. Additionally, we find that the velocity depends more strongly on the thickness of the myelin sheath than was suggested by previous theoretical studies. We further explain the slowing down and synchronisation of action potentials in ephaptically coupled fibres by their dynamic interaction. In summary, this study presents a solution to incorporate detailed axonal parameters into a whole-brain modelling framework.

With more and more data becoming available on white-matter tracts, the need arises to develop modelling frameworks that incorporate these data at the whole-brain level. This requires the development of efficient mathematical schemes to study parameter dependencies that can then be matched with data, in particular the speed of action potentials that cause delays between brain regions. Here, we develop a method that describes the formation of action potentials by threshold activated currents, often referred to as spike-diffuse-spike modelling. A particular focus of our study is the dependence of the speed of action potentials on structural parameters. We find that the diameter of axons and the thickness of the myelin sheath have a strong influence on the speed, whereas the length of myelinated segments and node of Ranvier length have a lesser effect. In addition to examining single axons, we demonstrate that action potentials between nearby axons can synchronise and slow down their propagation speed.

Model of Impedance Changes in Unmyelinated Nerve Fibers

OBJECTIVE

Currently, there is no imaging method that is able to distinguish the functional activity inside nerves. Such a method would be essential for understanding peripheral nerve physiology and would allow precise neuromodulation of organs these nerves supply. Electrical impedance tomography (EIT) is a method that produces images of electrical impedance change (dZ) of an object by injecting alternating current and recording surface voltages. It has been shown to be able to image fast activity in the brain and large peripheral nerves. To image inside small autonomic nerves, mostly containing unmyelinated fibers, it is necessary to maximize SNR and optimize the EIT parameters. An accurate model of the nerve is required to identify these optimal parameters as well as to validate data obtained in the experiments.

METHODS

In this study, we developed two three-dimensional models of unmyelinated fibers: Hodgkin-Huxley (HH) squid giant axon (single and multiple) and mammalian C-nociceptor. A coupling feedback system was incorporated into the models to simulate direct and alternating current application and simultaneously record external field during action potential propagation.

RESULTS

Parameters of the developed models were varied to study their influence on the recorded impedance changes; the optimal parameters were identified. The negative dZ was found to monotonically decrease with frequency for both HH and C fiber models, in accordance with the experimental data.

CONCLUSION AND SIGNIFICANCE

The accurate realistic model of unmyelinated nerve allows the optimization of EIT parameters and matches literature and experimental results.

Correlation of action potentials in adjacent neurons

A possible mechanism for the synchronization of action potential propagation along a bundle of neurons (ephaptic coupling) is considered. It is shown that this mechanism is similar to the salutatory conduction of the action potential between the nodes of Ranvier in myelinated axons. The proposed model allows us to estimate the scale of the correlation, i.e., the distance between neurons in the nervous tissue, wherein their synchronization becomes possible. The possibility for experimental verification of the proposed model of synchronization is discussed.

High-frequency oscillations are prominent in the extended amygdala

Previously, it was reported that various cortical and subcortical structures display high-frequency local field potential (LFP) oscillations in the 110- to 160-Hz range (HFOs), distinct from sharp-wave ripples. In the present study, we characterize HFOs in the extended amygdala. Rats were implanted with tetrode bundles in the bed nucleus of the stria terminalis (BNST), central amygdala (CeA), as well as adjacent regions (pallidum, caudate-putamen, and lateral septum). At all recorded sites, HFO power showed a systematic dependence on behavioral state: highest during quiet wakefulness, intermediate during paradoxical sleep, and lowest during active waking or slow-wave sleep. CO2 asphyxiation as well as anesthesia with isoflurane or urethane abolished HFOs. HFOs stood out relative to all other fast-frequency LFP components because they were highly coherent between distant sites of the extended amygdala, ipsi- and contralaterally. HFOs affected neuronal firing in two ways: firing rate could vary as a function of HFO power (rate modulation) or HFOs could entrain firing on a cycle-to-cycle basis (phase modulation). The incidence of phase-modulated neurons was about twice higher in BNST and CeA (20-40%) than in adjacent regions (≤8%). Among BNST and CeA neurons, many more were phase-modulated than rate-modulated, although about half of the latter were also phase-modulated. Overall, these results indicate that HFOs entrain the activity of a high proportion of neurons in the extended amygdala. A major challenge for future studies will be to identify the mechanisms supporting the high coherence of HFOs within and across hemispheres.

Tracking axonal action potential propagation on a high-density microelectrode array across hundreds of sites

Axons are traditionally considered stable transmission cables, but evidence of the regulation of action potential propagation demonstrates that axons may have more important roles. However, their small diameters render intracellular recordings challenging, and low-magnitude extracellular signals are difficult to detect and assign. Better experimental access to axonal function would help to advance this field. Here we report methods to electrically visualize action potential propagation and network topology in cortical neurons grown over custom arrays, which contain 11,011 microelectrodes and are fabricated using complementary metal oxide semiconductor technology. Any neuron lying on the array can be recorded at high spatio-temporal resolution, and simultaneously precisely stimulated with little artifact. We find substantial velocity differences occurring locally within single axons, suggesting that the temporal control of a neuron's output may contribute to neuronal information processing.

Lateral interactions in the outer retina

Lateral interactions in the outer retina, particularly negative feedback from horizontal cells to cones and direct feed-forward input from horizontal cells to bipolar cells, play a number of important roles in early visual processing, such as generating center-surround receptive fields that enhance spatial discrimination. These circuits may also contribute to post-receptoral light adaptation and the generation of color opponency. In this review, we examine the contributions of horizontal cell feedback and feed-forward pathways to early visual processing. We begin by reviewing the properties of bipolar cell receptive fields, especially with respect to modulation of the bipolar receptive field surround by the ambient light level and to the contribution of horizontal cells to the surround. We then review evidence for and against three proposed mechanisms for negative feedback from horizontal cells to cones: 1) GABA release by horizontal cells, 2) ephaptic modulation of the cone pedicle membrane potential generated by currents flowing through hemigap junctions in horizontal cell dendrites, and 3) modulation of cone calcium currents (I(Ca)) by changes in synaptic cleft proton levels. We also consider evidence for the presence of direct horizontal cell feed-forward input to bipolar cells and discuss a possible role for GABA at this synapse. We summarize proposed functions of horizontal cell feedback and feed-forward pathways. Finally, we examine the mechanisms and functions of two other forms of lateral interaction in the outer retina: negative feedback from horizontal cells to rods and positive feedback from horizontal cells to cones.

Action Potential Propagation Through Tissue Lacking Gap Junctions: Application to Engrafted Cells in Myocardial Infarcts

Engraftment of viable, electrically functional cells into a myocardial infarct as a method for restoring functionality is currently a topic of active research interest. Cells implanted in this way can form gap junction connectivity with each other, but often do not connect well with the surrounding tissue outside the infarct. Using a bidomain computer simulation model, we find that activation of these implanted cells by outside propagating action potentials is nevertheless possible, even if no gap junction connectivity to the surrounding tissue exists at all. The mechanism by which this action potential "tunneling" process occurs involves a current path that passes through both the intracellular and extracellular spaces, and is fundamentally spatially two-dimensional in nature. The typically convex boundary of the region occupied by these cells is found to greatly enhance the tunneling process, but unfortunately also hinders the ability of the activation of these cells to terminate reentrant waves propagating around the infarct.

Axon Physiology

Axons are generally considered as reliable transmission cables in which stable propagation occurs once an action potential is generated. Axon dysfunction occupies a central position in many inherited and acquired neurological disorders that affect both peripheral and central neurons. Recent findings suggest that the functional and computational repertoire of the axon is much richer than traditionally thought. Beyond classical axonal propagation, intrinsic voltage-gated ionic currents together with the geometrical properties of the axon determine several complex operations that not only control signal processing in brain circuits but also neuronal timing and synaptic efficacy. Recent evidence for the implication of these forms of axonal computation in the short-term dynamics of neuronal communication is discussed. Finally, we review how neuronal activity regulates both axon morphology and axonal function on a long-term time scale during development and adulthood.

Delay-induced destabilization of entrainment of nerve impulses on ephaptically coupled nerve fibers

We study the effect of delay on the synchronization of two nerve impulses traveling along two ephaptically coupled, unmyelinated nerve fibers. The system is modeled as a pair of delay-coupled Fitzhugh-Nagumo equations. A multiple-scale perturbation approach is used for the analysis of these equations in the limit of weak coupling. In the absence of delay, two pulses with identical speeds are shown to be entrained precisely. However, as the delay is increased beyond a critical value, we show that this precise entrainment becomes unstable. We make quantitative estimates for the actual values of delay at which this can occur in the case of squid giant axons and compare them with the relevant time scales involved.

Propagated repolarization of simulated action potentials in cardiac muscle and smooth muscle

BACKGROUND

Propagation of repolarization is a phenomenon that occurs in cardiac muscle. We wanted to test whether this phenomenon would also occur in our model of simulated action potentials (APs) of cardiac muscle (CM) and smooth muscle (SM) generated with the PSpice program.

METHODS

A linear chain of 5 cells was used, with intracellular stimulation of cell #1 for the antegrade propagation and of cell #5 for the retrograde propagation. The hyperpolarizing stimulus parameters applied for termination of the AP in cell #5 were varied over a wide range in order to generate strength / duration (S/D) curves. Because it was not possible to insert a second "black box" (voltage-controlled current source) into the basic units representing segments of excitable membrane that would allow the cells to respond to small hyperpolarizing voltages, gap-junction (g.j.) channels had to be inserted between the cells, represented by inserting a resistor (Rgj) across the four cell junctions.

RESULTS

Application of sufficient hyperpolarizing current to cell #5 to bring its membrane potential (Vm) to within the range of the sigmoidal curve of the Na+ conductance (CM) or Ca++ conductance (SM) terminated the AP in cell #5 in an all-or-none fashion. If there were no g.j. channels (Rgj = infinity), then only cell #5 repolarized to its stable resting potential (RP; -80 mV for CM and -55 mV for SM). The positive junctional cleft potential (VJC) produced only a small hyperpolarization of cell #4. However, if many g.j. channels were inserted, more hyperpolarizing current was required (for a constant duration) to repolarize cell #5, but repolarization then propagated into cells 4, 3, 2, and 1. When duration of the pulses was varied, a typical S/D curve, characteristic of excitable membranes, was produced. The chronaxie measured from the S/D curve was about 1.0 ms, similar to that obtained for muscle membranes.

CONCLUSIONS

These experiments demonstrate that normal antegrade propagation of excitation can occur in the complete absence of g.j. channels, and therefore no low-resistance pathways between cells, by the electric field (negative VJC) developed in the narrow junctional clefts. Because it was not possible to insert a second black-box into the basic units that would allow the cells to respond to small hyperpolarizing voltages, only cell #5 (the cell injected with hyperpolarizing pulses) repolarized in an all-or-none manner. But addition of many g.j. channels allowed repolarization to propagate in a retrograde direction over all 5 cells.

Gap-junction channels inhibit transverse propagation in cardiac muscle

The effect of adding many gap-junctions (g-j) channels between contiguous cells in a linear chain on transverse propagation between parallel chains was examined in a 5 × 5 model (5 parallel chains of 5 cells each) for cardiac muscle. The action potential upstrokes were simulated using the PSpice program for circuit analysis. Either a single cell was stimulated (cell A1) or the entire chain was stimulated simultaneously (A-chain). Transverse velocity was calculated from the total propagation time (TPT) from when the first AP crossed a V_m of -20 mV and the last AP crossed -20 mV. The number of g-j channels per junction was varied from zero to 100, 1,000 and 10,000 (R_gj of ∞, 100 MΩ, 10 MΩ, 1.0 MΩ, respectively). The longitudinal resistance of the interstitial fluid (ISF) space between the parallel chains (R_ol2) was varied between 200 KΩ (standard value) and 1.0, 5.0, and 10 MΩ. The higher the R_ol2 value, the tighter the packing of the chains. It was found that adding many g-j channels inhibited transverse propagation by blocking activation of all 5 chains, unless R_ol2 was greatly increased above the standard value of 200 KΩ. This was true for either method of stimulation. This was explained by, when there is strong longitudinal coupling between all 5 cells of a chain awaiting excitation, there must be more transfer energy (i.e., more current) to simultaneously excite all 5 cells of a chain.

Scaling laws for myelinated axons derived from an electrotonic core-conductor model

A macroscopic cable equation, which describes the passive linear ("electrotonic") response of a myelinated axon, was previously derived from a segmented cable equation using Keller's two-space homogenization method [Basser, PJ, Med. and Biol. Comput., 1993, Vol. 31, pp. S87-S92]. Here we use the space and length constants of this averaged cable equation to predict classical scaling laws that govern relationships among the inner and outer diameters of the axon's myelin sheath and the distance separating adjacent nodes of Ranvier. These laws are derived by maximizing the characteristic speed of an electrical disturbance along the axon, i.e., the ratio of the characteristic length and the characteristic time constants of the macroscopic cable, subject to the constraint that the nodal width is constant. Using this result, it is also possible to show that all myelinated axons are equally fault tolerant. No free parameters were used in this analysis; all variables and physical constants used in these calculations were taken from published experimental data.

Transverse propagation of action potentials between parallel chains of cardiac muscle and smooth muscle cells in PSpice simulations

BACKGROUND

We previously examined transverse propagation of action potentials between 2 and 3 parallel chain of cardiac muscle cells (CMC) simulated using the PSpice program. The present study was done to examine transverse propagation between 5 parallel chains in an expanded model of CMC and smooth muscle cells (SMC).

METHODS

Excitation was transmitted from cell to cell along a strand of 5 cells not connected by low-resistance tunnels (gap-junction connexons). The entire surface membrane of each cell fired nearly simultaneously, and nearly all the propagation time was spent at the cell junctions, the junctional delay time being about 0.3-0.5 ms (CMC) or 0.8-1.6 ms (SMC). A negative cleft potential (Vjc) develops in the narrow junctional clefts, whose magnitude depends on the radial cleft resistance (Rjc), which depolarizes the postjunctional membrane (post-JM) to threshold. Propagation velocity (theta) increased with amplitude of Vjc. Therefore, one mechanism for the transfer of excitation from one cell to the next is by the electric field (EF) that is generated in the junctional cleft when the pre-JM fires. In the present study, 5 parallel stands of 5 cells each (5 x 5 model) were used.

RESULTS

With electrical stimulation of the first cell of the first strand (cell A1), propagation rapidly spread down that chain and then jumped to the second strand (B chain), followed by jumping to the third, fourth, and fifth strands (C, D, E chains). The rapidity by which the parallel chains became activated depended on the longitudinal resistance of the narrow extracellular cleft between the parallel strands (Rol2); the higher the Rol2 resistance, the faster the theta. The transverse resistance of the cleft (Ror2) had almost no effect. Increasing Rjc decreases the total propagation time (TPT) over the 25-cell network. When the first cell of the third strand (cell C1) was stimulated, propagation spread down the C chain and jumped to the other two strands (B and D) nearly simultaneously.

CONCLUSIONS

Transverse propagation of excitation occurred at multiple points along the chain as longitudinal propagation was occurring, causing the APs in the contiguous chains to become bunched up. Transverse propagation was more erratic and labile in SMC compared to CMC. Transverse transmission of excitation did not require low-resistance connections between the chains, but instead depended on the value of Rol2. The tighter the packing of the chains facilitated transverse propagation.

Electrical interactions via the extracellular potential near cell bodies

Ephaptic interactions between a neuron and axons or dendrites passing by its cell body can be, in principle, more significant than ephaptic interactions among axons in a fiber tract. Extracellular action potentials outside axons are small in amplitude and spatially spread out, while they are larger in amplitude and much more spatially confined near cell bodies. We estimated the extracellular potentials associated with an action potential in a cortical pyramidal cell using standard one-dimensional cable theory and volume conductor theory. Their spatial and temporal pattern reveal much about the location and timing of currents in the cell, especially in combination with a known morphology, and simple experiments could resolve questions about spike initiation. From the extracellular potential we compute the ephaptically induced polarization in a nearby passive cable. The magnitude of this induced voltage can be several mV, does not spread electrotonically, and depends only weakly on the passive properties of the cable. We discuss their possible functional relevance.

Extracellular discontinuities in cardiac muscle: evidence for capillary effects on the action potential foot

It has become of fundamental importance to understand variations in the shape of the upstroke of the action potential in order to identify structural loading effects. One component of this goal is a detailed experimental analysis of the time course of the foot of the cardiac action potential (Vm foot) during propagation in different directions in anisotropic cardiac muscle. To this end, we performed phase-plane analysis of transmembrane action potentials during anisotropic propagation in adult working myocardium. The results showed that during longitudinal propagation there was initial slowing of Vm foot that resulted in deviations from a simple exponential; corollary changes occurred at numerous sites during transverse propagation. We hypothesized that the effect on Vm foot observed in the experimental data was created by the microscopic structure, especially the capillaries. This hypothesis predicts that the phase-plane trajectory of Vm foot will deviate from linearity in the presence of a high density of capillaries, and that a linear trajectory will occur in the absence of capillaries. Comparison of the results of Fast and Kléber (Circ Res. 1993;73:914-925) in a monolayer of neonatal cardiac myocytes, which is devoid of capillaries, and our results in newborn ventricular muscle, which is rich in capillaries, showed drastic differences in Vm foot as predicted. Because this comparison provided experimental support for the capillary hypothesis, we explored the underlying biophysical mechanisms due to interstitial electrical field effects, using a "2-domain" model of myocytes and capillaries separated by interstitial space. The model results show that a propagating interstitial electrical field induces an inward capacitive current in the inactive capillaries that causes a feedback effect on the active membrane (source) that slows the initial rise of its action potential. The results show unexpected mechanisms related to extracellular structural loading that may play a role in selected conduction disturbances, such as in a reperfused ischemic region surrounded by normal myocardium.

Motor unit estimates obtained using the new "MUESA" method

Motor unit number estimates were obtained from the extensor digitorum brevis and thenar muscles using a new method called MUESA. MUESA is distinguished from other estimation methods in the manner in which it deals with probabilistic motor unit activation, which is more commonly referred to as "alternation." Because of "alternation," incremental increases in the observed muscle potentials often cannot be interpreted in terms of the successive activation of single motor units. In the MUESA method, the nerve is subjected to a number of constant-intensity stimulus trains, and the resultant muscle response sequences are decomposed into their constituent motor unit action potentials. In general, if a stimulus train results in the probabilistic activation of n motor units, we can expect to see up to 2n different potentials, with each potential representing a unique combination of active and/or inactive motor units. If all 2n potentials are indeed observed, the decomposition of the observed potential sequence into its constituent motor unit action potentials is very straightforward. For the majority of the cases in which the number of observed potentials is not an integer power of 2, we have developed a novel decomposition method based on the analysis of the relative firing rates of the motor units.

Action potentials of curved nerves in finite limbs

Previous simulations of volume-conducted nerve-fiber action-potentials have modeled the limb as semi-infinite or circularly cylindrical, and the fibers as straight lines parallel to the limb surface. The geometry of actual nerves and limbs, however, can be considerably more complicated. This paper presents a general method for computing the potentials of fibers with arbitrary paths in arbitrary finite limbs. It involves computing the propagating point-source response (PPSR), which is the potential arising from a single point source (dipole or tripole) travelling along the fiber. The PPSR can be applied to fibers of different conduction velocities by simple dilation or compression. The method is illustrated for oblique and spiralling nerve fibers. Potentials from oblique fibers are shown to be different for orthodromic and antidromic propagation. Such results show that the straight-line models are not always adequate for nerves with anatomical amounts of curvature.

Linear algebraic transformations of the bidomain equations: implications for numerical methods

A mathematical framework is presented for the treatment of the bidomain equations used to model propagation in cardiac tissue. This framework is independent of the model used to represent membrane ionic currents and incorporates boundary conditions and other constraints. By representing the bidomain equations in the operator notation L phi = F, various algebraic transformations can be expressed as PLQ-1 psi = PF, where P and Q are linear operators. The authors show how previous work fits into this framework and discuss the implications of various transformation for numerical methods of solution. Although such transformations allow many choices of independent variable, these results emphasize the fundamental importance of the transmembrane potential.