Ion-concentration gradients induced by synaptic input increase the voltage depolarization in dendritic spines.

The vast majority of excitatory synaptic connections occur on dendritic spines. Due to their extremely small volume and spatial segregation from the dendrite, even moderate synaptic currents can significantly alter ionic concentrations. This results in chemical potential gradients between the dendrite and the spine head, leading to measurable electrical currents. In modeling electric signals in spines, different formalisms were previously used. While the cable equation is fundamental for understanding the electrical potential along dendrites, it only considers electrical currents as a result of gradients in electrical potential. The Poisson-Nernst-Planck (PNP) equations offer a more accurate description for spines by incorporating both electrical and chemical potential. However, solving PNP equations is computationally complex. In this work, diffusion currents are incorporated into the cable equation, leveraging an analogy between chemical and electrical potential. For simulating electric signals based on this extension of the cable equation, a straightforward numerical solver is introduced. The study demonstrates that this set of equations can be accurately solved using an explicit finite difference scheme. Through numerical simulations, this study unveils a previously unrecognized mechanism involving diffusion currents that amplify electric signals in spines. This discovery holds crucial implications for both numerical simulations and experimental studies focused on spine neck resistance and calcium signaling in dendritic spines.

Modeling Dendrites and Spatially-Distributed Neuronal Membrane Properties.

The first step toward understanding the brain is to learn how individual neurons process incoming signals, the vast majority of which arrive in their dendrites. Dendrites were first discovered at the beginning of the twentieth century and were characterized by great anatomical variability, both within and across species. Over the past years, a rich repertoire of active and passive dendritic mechanisms has been unveiled, which greatly influences their integrative power. Yet, our understanding of how dendrites compute remains limited, mainly because technological limitations make it difficult to record from dendrites directly and manipulate them. Computational modeling, on the other hand, is perfectly suited for this task. Biophysical models that account for the morphology as well as passive and active neuronal properties can explain a wide variety of experimental findings, shedding new light on how dendrites contribute to neuronal and circuit computations. This chapter aims to help the interested reader build biophysical models incorporating dendrites by detailing how their electrophysiological properties can be described using simple mathematical frameworks. We start by discussing the passive properties of dendrites and then give an overview of how active conductances can be incorporated, leading to realistic in silico replicas of biological neurons.

Nonsynaptic plasticity model of long-term memory engrams.

Using steady-state electrical properties of non-ohmic dendrite based on cable theory, we derive electrotonic potentials that do not change over time and are localized in space. We hypothesize that clusters of such stationary, local and permanent pulses are the electrical signatures of enduring memories which are imprinted through nonsynaptic plasticity, encoded through epigenetic mechanisms, and decoded through electrotonic processing. We further hypothesize how retrieval of an engram is made possible by integration of these permanently imprinted standing pulses in a neural circuit through neurotransmission in the extracellular space as part of conscious recall that acts as a guiding template in the reconsolidation of long-term memories through novelty characterized by uncertainty that arises when new fragments of memories reinstate an engram by way of nonsynaptic plasticity that permits its destabilization. Collectively, these findings seem to reinforce this hypothesis that electrotonic processing in non-ohmic dendrites yield insights into permanent electrical signatures that could reflect upon enduring memories as fragments of long-term memory engrams.

Biophysical constraints on lateral inhibition in the olfactory bulb.

The mitral cells (MCs) of the mammalian olfactory bulb (OB) constitute one of two populations of principal neurons (along with middle/deep tufted cells) that integrate afferent olfactory information with top-down inputs and intrinsic learning and deliver output to downstream olfactory areas. MC activity is regulated in part by inhibition from granule cells, which form reciprocal synapses with MCs along the extents of their lateral dendrites. However, with MC lateral dendrites reaching over 1.5 mm in length in rats, the roles of distal inhibitory synapses pose a quandary. Here, we systematically vary the properties of a MC model to assess the capacity of inhibitory synaptic inputs on lateral dendrites to influence afferent information flow through MCs. Simulations using passivized models with varying dendritic morphologies and synaptic properties demonstrated that, even with unrealistically favorable parameters, passive propagation fails to convey effective inhibitory signals to the soma from distal sources. Additional simulations using an active model exhibiting action potentials, subthreshold oscillations, and a dendritic morphology closely matched to experimental values further confirmed that distal synaptic inputs along the lateral dendrite could not exert physiologically relevant effects on MC spike timing at the soma. Larger synaptic conductances representative of multiple simultaneous inputs were not sufficient to compensate for the decline in signal with distance. Reciprocal synapses on distal MC lateral dendrites may instead serve to maintain a common fast oscillatory clock across the OB by delaying spike propagation within the lateral dendrites themselves.

Is realistic neuronal modeling realistic?

Scientific models are abstractions that aim to explain natural phenomena. A successful model shows how a complex phenomenon arises from relatively simple principles while preserving major physical or biological rules and predicting novel experiments. A model should not be a facsimile of reality; it is an aid for understanding it. Contrary to this basic premise, with the 21st century has come a surge in computational efforts to model biological processes in great detail. Here we discuss the oxymoronic, realistic modeling of single neurons. This rapidly advancing field is driven by the discovery that some neurons don't merely sum their inputs and fire if the sum exceeds some threshold. Thus researchers have asked what are the computational abilities of single neurons and attempted to give answers using realistic models. We briefly review the state of the art of compartmental modeling highlighting recent progress and intrinsic flaws. We then attempt to address two fundamental questions. Practically, can we realistically model single neurons? Philosophically, should we realistically model single neurons? We use layer 5 neocortical pyramidal neurons as a test case to examine these issues. We subject three publically available models of layer 5 pyramidal neurons to three simple computational challenges. Based on their performance and a partial survey of published models, we conclude that current compartmental models are ad hoc, unrealistic models functioning poorly once they are stretched beyond the specific problems for which they were designed. We then attempt to plot possible paths for generating realistic single neuron models.

Neuronal coupling by endogenous electric fields: cable theory and applications to coincidence detector neurons in the auditory brain stem.

The ongoing activity of neurons generates a spatially and time-varying field of extracellular voltage (Ve). This Ve field reflects population-level neural activity, but does it modulate neural dynamics and the function of neural circuits? We provide a cable theory framework to study how a bundle of model neurons generates Ve and how this Ve feeds back and influences membrane potential (Vm). We find that these "ephaptic interactions" are small but not negligible. The model neural population can generate Ve with millivolt-scale amplitude, and this Ve perturbs the Vm of "nearby" cables and effectively increases their electrotonic length. After using passive cable theory to systematically study ephaptic coupling, we explore a test case: the medial superior olive (MSO) in the auditory brain stem. The MSO is a possible locus of ephaptic interactions: sounds evoke large (millivolt scale)Vein vivo in this nucleus. The Ve response is thought to be generated by MSO neurons that perform a known neuronal computation with submillisecond temporal precision (coincidence detection to encode sound source location). Using a biophysically based model of MSO neurons, we find millivolt-scale ephaptic interactions consistent with the passive cable theory results. These subtle membrane potential perturbations induce changes in spike initiation threshold, spike time synchrony, and time difference sensitivity. These results suggest that ephaptic coupling may influence MSO function.

Dendrites impact the encoding capabilities of the axon.

This study highlights a new and powerful direct impact of the dendritic tree (the input region of neurons) on the encoding capability of the axon (the output region). We show that the size of the dendritic arbors (its impedance load) strongly modulates the shape of the action potential (AP) onset at the axon initial segment; it is accelerated in neurons with larger dendritic surface area. AP onset rapidness is key in determining the capability of the axonal spikes to encode (phase lock to) rapid changes in synaptic inputs. Hence, our findings imply that neurons with larger dendritic arbors have improved encoding capabilities. This "dendritic size effect" was explored both analytically as well as numerically, in simplified and detailed models of 3D reconstructed layer 2/3 cortical pyramidal cells of rats and humans. The cutoff frequency of spikes phase locking to modulated inputs increased from 100 to 200 Hz in pyramidal cells of young rats to 400-600 Hz in human cells. In the latter case, phase locking reached close to 1 KHz in in vivo-like conditions. This work highlights new and functionally profound cross talk between the dendritic tree and the axon initial segment, providing new understanding of neurons as sophisticated nonlinear input/output devices.

Portable conduction velocity experiments using earthworms for the college and high school neuroscience teaching laboratory.

The earthworm is ideal for studying action potential conduction velocity in a classroom setting, as its simple linear anatomy allows easy axon length measurements and the worm's sparse coding allows single action potentials to be easily identified. The earthworm has two giant fiber systems (lateral and medial) with different conduction velocities that can be easily measured by manipulating electrode placement and the tactile stimulus. Here, we present a portable and robust experimental setup that allows students to perform conduction velocity measurements within a 30-min to 1-h laboratory session. Our improvement over this well-known preparation is the combination of behaviorally relevant tactile stimuli (avoiding electrical stimulation) with the invention of minimal, low-cost, and portable equipment. We tested these experiments during workshops in both a high school and college classroom environment and found positive learning outcomes when we compared pre- and posttests taken by the students.

Basic neuron model electrical equivalent circuit: an undergraduate laboratory exercise.

We developed a hands-on laboratory exercise for undergraduate students in which they can build and manipulate a neuron equivalent circuit. This exercise uses electrical circuit components that resemble neuron components and are easy to construct. We describe the methods for creating the equivalent circuit and how to observe different neuron properties through altering the structure of the equivalent circuit. We explain how this hands-on laboratory activity allows for the better understanding of this fundamental neuroscience concept. At the conclusion of this laboratory exercise, undergraduate students will be able to apply the principles of Ohm's law, cable theory with regards to neurons, and understand the functions of resistance and capacitance in a neuron.

A general principle of dendritic constancy: A neuron's size- and shape-invariant excitability.

Reducing neuronal size results in less membrane and therefore lower input conductance. Smaller neurons are thus more excitable, as seen in their responses to somatic current injections. However, the impact of a neuron's size and shape on its voltage responses to dendritic synaptic activation is much less understood. Here we use analytical cable theory to predict voltage responses to distributed synaptic inputs in unbranched cables, showing that these are entirely independent of dendritic length. For a given synaptic density, neuronal responses depend only on the average dendritic diameter and intrinsic conductivity. This remains valid for a wide range of morphologies irrespective of their arborization complexity. Spiking models indicate that morphology-invariant numbers of spikes approximate the percentage of active synapses. In contrast to spike rate, spike times do depend on dendrite morphology. In summary, neuronal excitability in response to distributed synaptic inputs is largely unaffected by dendrite length or complexity.

A synaptic learning rule for exploiting nonlinear dendritic computation.

Information processing in the brain depends on the integration of synaptic input distributed throughout neuronal dendrites. Dendritic integration is a hierarchical process, proposed to be equivalent to integration by a multilayer network, potentially endowing single neurons with substantial computational power. However, whether neurons can learn to harness dendritic properties to realize this potential is unknown. Here, we develop a learning rule from dendritic cable theory and use it to investigate the processing capacity of a detailed pyramidal neuron model. We show that computations using spatial or temporal features of synaptic input patterns can be learned, and even synergistically combined, to solve a canonical nonlinear feature-binding problem. The voltage dependence of the learning rule drives coactive synapses to engage dendritic nonlinearities, whereas spike-timing dependence shapes the time course of subthreshold potentials. Dendritic input-output relationships can therefore be flexibly tuned through synaptic plasticity, allowing optimal implementation of nonlinear functions by single neurons.

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.

Theoretical relation between axon initial segment geometry and excitability.

In most vertebrate neurons, action potentials are triggered at the distal end of the axon initial segment (AIS). Both position and length of the AIS vary across and within neuron types, with activity, development and pathology. What is the impact of AIS geometry on excitability? Direct empirical assessment has proven difficult because of the many potential confounding factors. Here, we carried a principled theoretical analysis to answer this question. We provide a simple formula relating AIS geometry and sodium conductance density to the somatic voltage threshold. A distal shift of the AIS normally produces a (modest) increase in excitability, but we explain how this pattern can reverse if a hyperpolarizing current is present at the AIS, due to resistive coupling with the soma. This work provides a theoretical tool to assess the significance of structural AIS plasticity for electrical function.

A mechanoelectrical coupling model of neurons under stretching.

Neurons are situated in a microenvironment composed of various mechanical cues, where stretching is thought to have a major impact on neurons, resulting in microstructural changes in neural tissue and further leading to abnormal electrophysiological function. In spite of significant experimental efforts, the underlying mechanism remains elusive, more works are needed to provide a detailed description of the process that leads to the observed phenomena. Here, we developed a mechanoelectrical coupling model of central neurons under stretching and specially considered the plastic deformation of neurons. With the model, we showed that the increasing axial strain induces a decreased membrane action potential and a more frequent neuronal firing, which agree well with experimental observations reported in the literature. The simulation results also showed a faster electrophysiological signal conduction. Our model provides a reference for the prediction and regulation of neuronal function under simplified conditions of mechanical loadings.

Evidence for Quantized Magnetic Flux in an Axon.

In December of 2018 I published my consolidated findings of a closed-form description of propagated signaling phenomena in the membrane of an axon [1]. Those results demonstrate how intracellular conductance, the thermodynamics of magnetization, and current modulation, function together in generating an action potential in a unified differential equation. At present, I report on a subsequent finding within this model. Namely, evidence of quantized magnetic flux Φ0 in an axon.

Uniaxial strain of cultured mouse and rat cardiomyocyte strands slows conduction more when its axis is parallel to impulse propagation than when it is perpendicular.

AIM: Cardiac tissue deformation can modify tissue resistance, membrane capacitance and ion currents and hence cause arrhythmogenic slow conduction. Our aim was to investigate whether uniaxial strain causes different changes in conduction velocity (θ) when the principal strain axis is parallel vs perpendicular to impulse propagation. METHODS: Cardiomyocyte strands were cultured on stretchable custom microelectrode arrays, and θ was determined during steady-state pacing. Uniaxial strain (5%) with principal axis parallel (orthodromic) or perpendicular (paradromic) to propagation was applied for 1 minute and controlled by imaging a grid of markers. The results were analysed in terms of cable theory. RESULTS: Both types of strain induced immediate changes of θ upon application and release. In material coordinates, orthodromic strain decreased θ significantly more (P < .001) than paradromic strain (2.2 ± 0.5% vs 1.0 ± 0.2% in n = 8 mouse cardiomyocyte cultures, 2.3 ± 0.4% vs 0.9 ± 0.5% in n = 4 rat cardiomyocyte cultures, respectively). The larger effect of orthodromic strain can be explained by the increase in axial myoplasmic resistance, which is not altered by paradromic strain. Thus, changes in tissue resistance substantially contributed to the changes of θ during strain, in addition to other influences (eg stretch-activated channels). Besides these immediate effects, the application of strain also consistently initiated a slow progressive decrease in θ and a slow recovery of θ upon release. CONCLUSION: Changes in cardiac conduction velocity caused by acute stretch do not only depend on the magnitude of strain but also on its orientation relative to impulse propagation. This dependence is due to different effects on tissue resistance.

A Single Differential Equation Description of Membrane Properties Underlying the Action Potential and the Axon Electric Field.

In a succession of articles published over 65 years ago, Sir Alan Lloyd Hodgkin and Sir Andrew Fielding Huxley established what now forms our physical understanding of excitation in nerve, and how the axon conducts the action potential. They uniquely quantified the movement of ions in the nerve cell during the action potential, and demonstrated that the action potential is the result of a depolarizing event across the cell membrane. They confirmed that a complete depolarization event is followed by an abrupt increase in voltage that propagates longitudinally along the axon, accompanied by considerable increases in membrane conductance. In an elegant theoretical framework, they rigorously described fundamental properties of the Na+ and K+ conductances intrinsic to the action potential. Notwithstanding the elegance of Hodgkin and Huxley's incisive and explicative series of discoveries, their model is mathematically complex, relies on no small number of stochastic factors, and has no analytical solution. Solving for the membrane action potential and the ionic currents requires integrations approximated using numerical methods. In this article I present an analytical formalism of the nerve action potential, Vm and that of the accompanying cell membrane electric field, Em . To conclude, I present a novel description of Vm in terms of a single, nonlinear differential equation. This is an original stand-alone article: the major contribution is the latter, and how this description coincides with the cell membrane electric field. This work has necessitated unifying information from two preceding papers [1,2], each being concerned with the development of closed-form descriptions of the nerve action potential, Vm .

The role of negative conductances in neuronal subthreshold properties and synaptic integration.

Based on passive cable theory, an increase in membrane conductance produces a decrease in the membrane time constant and input resistance. Unlike the classical leak currents, voltage-dependent currents have a nonlinear behavior which can create regions of negative conductance, despite the increase in membrane conductance (permeability). This negative conductance opposes the effects of the passive membrane conductance on the membrane input resistance and time constant, increasing their values and thereby substantially affecting the amplitude and time course of postsynaptic potentials at the voltage range of the negative conductance. This paradoxical effect has been described for three types of voltage-dependent inward currents: persistent sodium currents, L- and T-type calcium currents and ligand-gated glutamatergic N-methyl-D-aspartate currents. In this review, we describe the impact of the creation of a negative conductance region by these currents on neuronal membrane properties and synaptic integration. We also discuss recent contributions of the quasi-active cable approximation, an extension of the passive cable theory that includes voltage-dependent currents, and its effects on neuronal subthreshold properties.

Remodeling of cardiac passive electrical properties and susceptibility to ventricular and atrial arrhythmias.

Coordinated electrical activation of the heart is essential for the maintenance of a regular cardiac rhythm and effective contractions. Action potentials spread from one cell to the next via gap junction channels. Because of the elongated shape of cardiomyocytes, longitudinal resistivity is lower than transverse resistivity causing electrical anisotropy. Moreover, non-uniformity is created by clustering of gap junction channels at cell poles and by non-excitable structures such as collagenous strands, vessels or fibroblasts. Structural changes in cardiac disease often affect passive electrical properties by increasing non-uniformity and altering anisotropy. This disturbs normal electrical impulse propagation and is, consequently, a substrate for arrhythmia. However, to investigate how these structural changes lead to arrhythmias remains a challenge. One important mechanism, which may both cause and prevent arrhythmia, is the mismatch between current sources and sinks. Propagation of the electrical impulse requires a sufficient source of depolarizing current. In the case of a mismatch, the activated tissue (source) is not able to deliver enough depolarizing current to trigger an action potential in the non-activated tissue (sink). This eventually leads to conduction block. It has been suggested that in this situation a balanced geometrical distribution of gap junctions and reduced gap junction conductance may allow successful propagation. In contrast, source-sink mismatch can prevent spontaneous arrhythmogenic activity in a small number of cells from spreading over the ventricle, especially if gap junction conductance is enhanced. Beside gap junctions, cell geometry and non-cellular structures strongly modulate arrhythmogenic mechanisms. The present review elucidates these and other implications of passive electrical properties for cardiac rhythm and arrhythmogenesis.

A novel mechanism for switching a neural system from one state to another.

An animal's ability to rapidly adjust to new conditions is essential to its survival. The nervous system, then, must be built with the flexibility to adjust, or shift, its processing capabilities on the fly. To understand how this flexibility comes about, we tracked a well-known behavioral shift, a visual integration shift, down to its underlying circuitry, and found that it is produced by a novel mechanism - a change in gap junction coupling that can turn a cell class on and off. The results showed that the turning on and off of a cell class shifted the circuit's behavior from one state to another, and, likewise, the animal's behavior. The widespread presence of similar gap junction-coupled networks in the brain suggests that this mechanism may underlie other behavioral shifts as well.