Stimulation-induced ectopicity and propagation windows in model damaged axons

Unraveling the dynamics of firing patterns for neurons with impairment of sodium channels

Various factors such as mechanical trauma, chemical trauma, local ischemia, and inflammation can impair voltage-gated sodium channels (Nav) in neurons. These impairments lead to a distinctive leftward shift in the activation and inactivation curves of voltage-gated sodium channels. The resulting sodium channel impairments in neurons are known to affect firing patterns, which play a significant role in neuronal activities within the nervous system. However, the underlying dynamic mechanism for the emergence of these firing patterns remains unclear. In this study, we systematically investigated the effects of sodium channel dysfunction on individual neuronal dynamics and firing patterns. By employing codimension-1 bifurcation analysis, we revealed the underlying dynamical mechanism responsible for the generation of different firing patterns. Additionally, through codimension-2 bifurcation analysis, we theoretically determined the distribution of firing patterns on different parameter planes. Our results indicate that the firing patterns of impaired neurons are regulated by multiple parameters, with firing pattern transitions caused by the degree of sodium channel impairment being more diverse than those caused by the ratio of impaired sodium channel and current. Furthermore, we observed that the firing pattern of tonic firing is more likely to be the norm in impaired sodium channel neurons, providing valuable insights into the signaling of impaired neurons. Overall, our findings highlight the intricate relationships among sodium channel impairments, neuronal dynamics, and firing patterns, shedding light on the impact of disruptions in ion concentration gradients on neuronal function.

Slow-gamma frequencies are optimally guarded against effects of neurodegenerative diseases and traumatic brain injuries

We introduce a computational model for the cellular level effects of firing rate filtering due to the major forms of neuronal injury, including demyelination and axonal swellings. Based upon experimental and computational observations, we posit simple phenomenological input/output rules describing spike train distortions and demonstrate that slow-gamma frequencies in the 38-41 Hz range emerge as the most robust to injury. Our signal-processing model allows us to derive firing rate filters at the cellular level for impaired neural activity with minimal assumptions. Specifically, we model eight experimentally observed spike train transformations by discrete-time filters, including those associated with increasing refractoriness and intermittent blockage. Continuous counterparts for the filters are also obtained by approximating neuronal firing rates from spike trains convolved with causal and Gaussian kernels. The proposed signal processing framework, which is robust to model parameter calibration, is an abstraction of the major cellular-level pathologies associated with neurodegenerative diseases and traumatic brain injuries that affect spike train propagation and impair neuronal network functionality. Our filters are well aligned with the spectrum of dynamic memory fields including working memory, visual consciousness, and other higher cognitive functions that operate in a frequency band that is - at a single cell level - optimally guarded against common types of pathological effects. In contrast, higher-frequency neural encoding, such as is observed with short-term memory, are susceptible to neurodegeneration and injury.

Linking demyelination to compound action potential dispersion with a spike-diffuse-spike approach

To establish and exploit novel biomarkers of demyelinating diseases requires a mechanistic understanding of axonal propagation. Here, we present a novel computational framework called the stochastic spike-diffuse-spike (SSDS) model for assessing the effects of demyelination on axonal transmission. It models transmission through nodal and internodal compartments with two types of operations: a stochastic integrate-and-fire operation captures nodal excitability and a linear filtering operation describes internodal propagation. The effects of demyelinated segments on the probability of transmission, transmission delay and spike time jitter are explored. We argue that demyelination-induced impedance mismatch prevents propagation mostly when the action potential leaves a demyelinated region, not when it enters a demyelinated region. In addition, we model sodium channel remodeling as a homeostatic control of nodal excitability. We find that the effects of mild demyelination on transmission probability and delay can be largely counterbalanced by an increase in excitability at the nodes surrounding the demyelination. The spike timing jitter, however, reflects the level of demyelination whether excitability is fixed or is allowed to change in compensation. This jitter can accumulate over long axons and leads to a broadening of the compound action potential, linking microscopic defects to a mesoscopic observable. Our findings articulate why action potential jitter and compound action potential dispersion can serve as potential markers of weak and sporadic demyelination.

Cooling reverses pathological bifurcations to spontaneous firing caused by mild traumatic injury

Mild traumatic injury can modify the key sodium (Na⁺) current underlying the excitability of neurons. It causes the activation and inactivation properties of this current to become shifted to more negative trans-membrane voltages. This so-called coupled left shift (CLS) leads to a chronic influx of Na⁺ into the cell that eventually causes spontaneous or "ectopic" firing along the axon, even in the absence of stimuli. The bifurcations underlying this enhanced excitability have been worked out in full ionic models of this effect. Here, we present computational evidence that increased temperature T can exacerbate this pathological state. Conversely, and perhaps of clinical relevance, mild cooling is shown to move the naturally quiescent cell further away from the threshold of ectopic behavior. The origin of this stabilization-by-cooling effect is analyzed by knocking in and knocking out, one at a time, various processes thought to be T-dependent. The T-dependence of the Na⁺ current, quantified by its Q _10-Na factor, has the biggest impact on the threshold, followed by Q _10-pump of the sodium-potassium exchanger. Below the ectopic boundary, the steady state for the gating variables and the resting potential are not modified by temperature, since our model separately tallies the Na⁺ and K⁺ ions including their separate leaks through the pump. When only the gating kinetics are considered, cooling is detrimental, but in the full T-dependent model, it is beneficial because the other processes dominate. Cooling decreases the pump's activity, and since the pump hyperpolarizes, less hyperpolarization should lead to more excitability and ectopic behavior. But actually the opposite happens in the full model because decreased pump activity leads to smaller gradients of Na⁺ and K⁺, which in turn decreases the driving force of the Na⁺ current.

Cytotoxic Swelling of Sick Excitable Cells - Impaired Ion Homeostasis and Membrane Tension Homeostasis in Muscle and Neuron

When they become simultaneously leaky to both Na⁺ and Cl^-, excitable cells are vulnerable to potentially lethal cytotoxic swelling. Swelling ensues in spite of an isosmotic milieu because the entering ions add osmolytes to the cytoplasm's high concentration of impermeant anionic osmolytes. An influx of osmotically-obliged water is unavoidable. A cell that cannot stanch at least one the leaks will succumb to death by Donnan effect. "Sick excitable cells" are those injured through ischemia, trauma, inflammation, hyperactivity, genetically-impaired membrane skeletons and other insults, all of which foster bleb-damage to regions of the plasma membrane. Nav channels resident in damaged membrane exhibit left-shifted kinetics; the corresponding Nav window conductance constitutes a Na⁺-leak. In cortical neurons, sustained depolarization to ∼-20mV elicits a sustained lethal gCl. Underlying V_rest in skeletal muscle is a constitutively active gCl; not surprisingly therefore, dystrophic muscle fibers, which are prone to bleb damage and which exhibit Nav-leak and Na⁺-overload, are prone to cytotoxic swelling. To restore viability in cytotoxically swelling neurons and muscle, the imperative of fully functional ion homeostasis is well-recognized. However, as emphasized here, in a healthy excitable cell, fully functional membrane tension homeostasis is also imperative. ATPase-pumps keep plasma membrane batteries charged, and ATPase-motor proteins maintain membrane tone. In sick excitable cells, neither condition prevails.

Persistent sodium current modulates axonal excitability in CA1 pyramidal neurons

Axonal excitability is an important determinant for the accuracy, direction, and velocity of neuronal signaling. The mechanisms underlying spike generation in the axonal initial segment and transmitter release from presynaptic terminals have been intensely studied and revealed a role for several specific ionic conductances, including the persistent sodium current (I_NaP ). Recent evidence indicates that action potentials can also be generated at remote locations along the axonal fiber, giving rise to ectopic action potentials during physiological states (e.g., fast network oscillations) or in pathological situations (e.g., following demyelination). Here, we investigated how ectopic axonal excitability of mouse hippocampal CA1 pyramidal neurons is regulated by I_NaP . Recordings of field potentials and intracellular voltage in brain slices revealed that electrically evoked antidromic spikes were readily suppressed by two different blockers of I_NaP , riluzole and phenytoin. The effect was mediated by a reduction of the probability of ectopic spike generation while latency was unaffected. Interestingly, the contribution of I_NaP to excitability was much more pronounced in axonal branches heading toward the entorhinal cortex compared with the opposite fiber direction toward fimbria. Thus, excitability of distal CA1 pyramidal cell axons is affected by persistent sodium currents in a direction-selective manner. This mechanism may be of importance for ectopic spike generation in oscillating network states as well as in pathological situations.

Ionic mechanisms underlying history-dependence of conduction delay in an unmyelinated axon

Axonal conduction velocity can change substantially during ongoing activity, thus modifying spike interval structures and, potentially, temporal coding. We used a biophysical model to unmask mechanisms underlying the history-dependence of conduction. The model replicates activity in the unmyelinated axon of the crustacean stomatogastric pyloric dilator neuron. At the timescale of a single burst, conduction delay has a non-monotonic relationship with instantaneous frequency, which depends on the gating rates of the fast voltage-gated Na⁺ current. At the slower timescale of minutes, the mean value and variability of conduction delay increase. These effects are because of hyperpolarization of the baseline membrane potential by the Na⁺/K⁺ pump, balanced by an h-current, both of which affect the gating of the Na⁺ current. We explore the mechanisms of history-dependence of conduction delay in axons and develop an empirical equation that accurately predicts this history-dependence, both in the model and in experimental measurements.

Reaction time impairments in decision-making networks as a diagnostic marker for traumatic brain injuries and neurological diseases

The presence of diffuse Focal Axonal Swellings (FAS) is a hallmark cellular feature in many neurological diseases and traumatic brain injury. Among other things, the FAS have a significant impact on spike-train encodings that propagate through the affected neurons, leading to compromised signal processing on a neuronal network level. This work merges, for the first time, three fields of study: (i) signal processing in excitatory-inhibitory (EI) networks of neurons via population codes, (ii) decision-making theory driven by the production of evidence from stimulus, and (iii) compromised spike-train propagation through FAS. As such, we demonstrate a mathematical architecture capable of characterizing compromised decision-making driven by cellular mechanisms. The computational model also leads to several novel predictions and diagnostics for understanding injury level and cognitive deficits, including a key finding that decision-making reaction times, rather than accuracy, are indicative of network level damage. The results have a number of translational implications, including that the level of network damage can be characterized by the reaction times in simple cognitive and motor tests.

Calculating the Consequences of Left-Shifted Nav Channel Activity in Sick Excitable Cells

Two features common to diverse sick excitable cells are "leaky" Nav channels and bleb damage-damaged membranes. The bleb damage, we have argued, causes a channel kinetics based "leakiness." Recombinant (node of Ranvier type) Nav1.6 channels voltage-clamped in mechanically-blebbed cell-attached patches undergo a damage intensity dependent kinetic change. Specifically, they experience a coupled hyperpolarizing (left) shift of the activation and inactivation processes. The biophysical observations on Nav1.6 currents formed the basis of Nav-Coupled Left Shift (Nav-CLS) theory. Node of Ranvier excitability can be modeled with Nav-CLS imposed at varying LS intensities and with varying fractions of total nodal membrane affected. Mild damage from which sick excitable cells might recover is of most interest pathologically. Accordingly, Na⁺/K⁺ ATPase (pump) activity was included in the modeling. As we described more fully in our other recent reviews, Nav-CLS in nodes with pumps proves sufficient to predict many of the pathological excitability phenomena reported for sick excitable cells. This review explains how the model came about and outlines how we have used it. Briefly, we direct the reader to studies in which Nav-CLS is being implemented in larger scale models of damaged excitable tissue. For those who might find it useful for teaching or research purposes, we coded the Nav-CLS/node of Ranvier model (with pumps) in NEURON. We include, here, the resulting "Regimes" plot of classes of excitability dysfunction.

Improved Simulation of Electrodiffusion in the Node of Ranvier by Mesh Adaptation

In neural structures with complex geometries, numerical resolution of the Poisson-Nernst-Planck (PNP) equations is necessary to accurately model electrodiffusion. This formalism allows one to describe ionic concentrations and the electric field (even away from the membrane) with arbitrary spatial and temporal resolution which is impossible to achieve with models relying on cable theory. However, solving the PNP equations on complex geometries involves handling intricate numerical difficulties related either to the spatial discretization, temporal discretization or the resolution of the linearized systems, often requiring large computational resources which have limited the use of this approach. In the present paper, we investigate the best ways to use the finite elements method (FEM) to solve the PNP equations on domains with discontinuous properties (such as occur at the membrane-cytoplasm interface). 1) Using a simple 2D geometry to allow comparison with analytical solution, we show that mesh adaptation is a very (if not the most) efficient way to obtain accurate solutions while limiting the computational efforts, 2) We use mesh adaptation in a 3D model of a node of Ranvier to reveal details of the solution which are nearly impossible to resolve with other modelling techniques. For instance, we exhibit a non linear distribution of the electric potential within the membrane due to the non uniform width of the myelin and investigate its impact on the spatial profile of the electric field in the Debye layer.

Nav Channels in Damaged Membranes

Sick excitable cells (ie, Nav channel-expressing cells injured by trauma, ischemia, inflammatory, and other conditions) typically exhibit "acquired sodium channelopathies" which, we argue, reflect bleb-damaged membranes rendering their Nav channels "leaky." The situation is excitotoxic because untreated Nav leak exacerbates bleb damage. Fast Nav inactivation (a voltage-independent process) is so tightly coupled, kinetically speaking, to the inherently voltage-dependent process of fast activation that when bleb damage accelerates and thus left-shifts macroscopic fast activation, fast inactivation accelerates to the same extent. The coupled g(V) and availability(V) processes and their window conductance regions consequently left-shift by the same number of millivolts. These damage-induced hyperpolarizing shifts, whose magnitude increases with damage intensity, are called coupled left shift (CLS). Based on past work and modeling, we discuss how to test for Nav-CLS, emphasizing the virtue of sawtooth ramp clamp. We explain that it is the inherent mechanosensitivity of Nav activation that underlies Nav-CLS. Using modeling of excitability, we show the known process of Nav-CLS is sufficient to predict a wide variety of "sick excitable cell" phenomena, from hyperexcitability through to depolarizing block. When living cells are mimicked by inclusion of pumps, mild Nav-CLS produces a wide array of burst phenomena and subthreshold oscillations. Dynamical analysis of mild damage scenarios shows how these phenomena reflect changes in spike thresholds as the pumps try to counteract the leaky Nav channels. Smart Nav inhibitors designed for sick excitable cells would target bleb-damaged membrane, buying time for cell-mediated removal or repair of Nav-bearing membrane that has become bleb-damaged (ie, detached from the cytoskeleton).

Perinodal glial swelling mitigates axonal degradation in a model of axonal injury

Mild traumatic brain injury (mTBI) has been associated with the damage to myelinated axons in white matter tracts. Animal models and in vitro studies suggest that axonal degradation develops during a latent period following a traumatic event. This delay has been attributed to slowly developing axonal membrane depolarization that is initiated by injury-induced ionic imbalance and in turn, leads to the activation of Ca(2+) proteases via pathological accumulation of Ca(2+). However, the mechanisms mitigating the transition to axonal degradation after injury remain elusive. We addressed this question in a detailed biophysical model of axonal injury that incorporated ion exchange and glial swelling mechanisms. We show that glial swelling, which often co-occurs with mTBI, promotes axonal survival by regulating extracellular K(+) dynamics, extending the range of injury parameters in which axons exhibit stable membrane potential postinjury. In addition, glial swelling was instrumental in reducing axonal sensitivity to repetitive stretch injury that occurred several minutes following the first one. Results of this study suggest that acute post-traumatic swelling of perinodal astrocytes helps prevent or postpone axonal degradation by maintaining physiologically relevant levels of extracellular K(+).

From Squid to Mammals with the HH Model through the Nav Channels' Half-Activation-Voltage Parameter

The model family analyzed in this work stems from the classical Hodgkin-Huxley model (HHM). for a single-compartment (space-clamp) and continuous variation of the voltage-gated sodium channels (Na_v) half-activation-voltage parameter ΔV_1/2, which controls the window of sodium-influx currents. Unlike the baseline HHM, its parametric extension exhibits a richer multitude of dynamic regimes, such as multiple fixed points (FP’s), bi- and multi-stability (coexistence of FP’s and/or periodic orbits). Such diversity correlates with a number of functional properties of excitable neural tissue, such as the capacity or not to evoke an action potential (AP) from the resting state, by applying a minimal absolute rheobase current amplitude. The utility of the HHM rooted in the giant squid for the descriptions of the mammalian nervous system is of topical interest. We conclude that the model’s fundamental principles are still valid (up to using appropriate parameter values) for warmer-blooded species, without a pressing need for a substantial revision of the mathematical formulation. We demonstrate clearly that the continuous variation of the ΔV_1/2 parameter comes close to being equivalent with recent HHM ‘optimizations’. The neural dynamics phenomena described here are nontrivial. The model family analyzed in this work contains the classical HHM as a special case. The validity and applicability of the HHM to mammalian neurons can be achieved by picking the appropriate ΔV_1/2 parameter in a significantly broad range of values. For such large variations, in contrast to the classical HHM, the h and n gates’ dynamics may be uncoupled - i.e. the n gates may no longer be considered in mere linear correspondence to the h gates. ΔV_1/2variation leads to a multitude of dynamic regimes—e.g. models with either 1 fixed point (FP) or with 3 FP’s. These may also coexist with stable and/or unstable periodic orbits. Hence, depending on the initial conditions, the system may behave as either purely excitable or as an oscillator. ΔV_1/2 variation leads to significant changes in the metabolic efficiency of an action potential (AP). Lower ΔV_1/2 values yield a larger range of AP response frequencies, and hence provide for more flexible neural coding. Such lower values also contribute to faster AP conduction velocities along neural fibers of otherwise comparable-diameter. The 3 FP case brings about an absolute rheobase current. In comparison in the classical HHM the rheobase current is only relative - i.e. excitability is lost after a finite amount of elapsed stimulation time. Lower ΔV_1/2 values translate in lower threshold currents from the resting state.