Relating ion channel expression, bifurcation structure, and diverse firing patterns in a model of an identified motor neuron

The Neurobehavioral State hypothesis

Since the early attempts to understand the brain made by Greek philosophers more than 2000 years ago, one of the main questions in neuroscience has been how the brain perceives all the stimuli in the environment and uses this information to implement a response. Recent hypotheses of the neural code rely on the existence of an ideal observer, whether on specific areas of the cerebral cortex or distributed network composed of cortical and subcortical elements. The Neurobehavioral State hypothesis stipulates that neurons are in a quasi-stable state due to the dynamic interaction of their molecular components. This increases their computational capabilities and electrophysiological behavior further than a binary active/inactive state. Together, neuronal populations across the brain learn to identify and associate internal and external stimuli with actions and emotions. Furthermore, such associations can be stored through the regulation of neuronal components as new quasi-stable states. Using this framework, behavior arises as the result of the dynamic interaction between internal and external stimuli together with previously established quasi-stable states that delineate the behavioral response. Finally, the Neurobehavioral State hypothesis is firmly grounded on present evidence of the complex dynamics within the brain, from the molecular to the network level, and avoids the need for a central observer by proposing the brain configures itself through experience-driven associations.

A biophysical minimal model to investigate age-related changes in CA1 pyramidal cell electrical activity

Aging is a physiological process that is still poorly understood, especially with respect to effects on the brain. There are open questions about aging that are difficult to answer with an experimental approach. Underlying challenges include the difficulty of recording in vivo single cell and network activity simultaneously with submillisecond resolution, and brain compensatory mechanisms triggered by genetic, pharmacologic, or behavioral manipulations. Mathematical modeling can help address some of these questions by allowing us to fix parameters that cannot be controlled experimentally and investigate neural activity under different conditions. We present a biophysical minimal model of CA1 pyramidal cells (PCs) based on general expressions for transmembrane ion transport derived from thermodynamical principles. The model allows directly varying the contribution of ion channels by changing their number. By analyzing the dynamics of the model, we find parameter ranges that reproduce the variability in electrical activity seen in PCs. In addition, increasing the L-type Ca2+ channel expression in the model reproduces age-related changes in electrical activity that are qualitatively and quantitatively similar to those observed in PCs from aged animals. We also make predictions about age-related changes in PC bursting activity that, to our knowledge, have not been reported previously. We conclude that the model's biophysical nature, flexibility, and computational simplicity make it a potentially powerful complement to experimental studies of aging.

Biological complexity facilitates tuning of the neuronal parameter space

The electrical and computational properties of neurons in our brains are determined by a rich repertoire of membrane-spanning ion channels and elaborate dendritic trees. However, the precise reason for this inherent complexity remains unknown, given that simpler models with fewer ion channels are also able to functionally reproduce the behaviour of some neurons. Here, we stochastically varied the ion channel densities of a biophysically detailed dentate gyrus granule cell model to produce a large population of putative granule cells, comparing those with all 15 original ion channels to their reduced but functional counterparts containing only 5 ion channels. Strikingly, valid parameter combinations in the full models were dramatically more frequent at -6% vs. -1% in the simpler model. The full models were also more stable in the face of perturbations to channel expression levels. Scaling up the numbers of ion channels artificially in the reduced models recovered these advantages confirming the key contribution of the actual number of ion channel types. We conclude that the diversity of ion channels gives a neuron greater flexibility and robustness to achieve a target excitability.

Distinctions among electroconvulsion- and proconvulsant-induced seizure discharges and native motor patterns during flight and grooming: quantitative spike pattern analysis in Drosophila flight muscles

In Drosophila, high-frequency electrical stimulation across the brain triggers a highly stereotypic repertoire of spasms. These electroconvulsive seizures (ECS) manifest as distinctive spiking discharges across the nervous system and can be stably assessed throughout the seizure repertoire in the large indirect flight muscles dorsal longitudinal muscles (DLMs) to characterize modifications in seizure-prone mutants. However, the relationships between ECS-spike patterns and native motor programs, including flight and grooming, are not known and their similarities and distinctions remain to be characterized. We employed quantitative spike pattern analyses for the three motor patterns including: (1) overall firing frequency, (2) spike timing between contralateral fibers, and (3) short-term variability in spike interval regularity (CV₂) and instantaneous firing frequency (ISI^-1). This base-line information from wild-type (WT) flies facilitated quantitative characterization of mutational effects of major neurotransmitter systems: excitatory cholinergic (Cha), inhibitory GABAergic (Rdl) and electrical (ShakB) synaptic transmission. The results provide an initial glimpse on the vulnerability of individual motor patterns to different perturbations. We found marked alterations of ECS discharge spike patterns in terms of either seizure threshold, spike frequency or spiking regularity. In contrast, no gross alterations during grooming and a small but noticeable reduction of firing frequency during Rdl mutant flight were found, suggesting a role for GABAergic modulation of flight motor programs. Picrotoxin (PTX), a known pro-convulsant that inhibits GABA_A receptors, induced DLM spike patterns that displayed some features, e.g. left-right coordination and ISI^-1 range, that could be found in flight or grooming, but distinct from ECS discharges. These quantitative techniques may be employed to reveal overlooked relationships among aberrant motor patterns as well as their links to native motor programs.

A thermodynamic description for physiological transmembrane transport

A general formulation for both passive and active transmembrane transport is derived from basic thermodynamical principles. The derivation takes into account the energy required for the motion of molecules across membranes and includes the possibility of modeling asymmetric flow. Transmembrane currents can then be described by the general model in the case of electrogenic flow. As it is desirable in new models, it is possible to derive other well-known expressions for transmembrane currents as particular cases of the general formulation. For instance, the conductance-based formulation for current turns out to be a linear approximation of the general formula for current. Also, under suitable assumptions, other formulas for current based on electrodiffusion, like the constant field approximation by Goldman, can be recovered from the general formulation. The applicability of the general formulations is illustrated first with fits to existing data, and after, with models of transmembrane potential dynamics for pacemaking cardiocytes and neurons. The general formulations presented here provide a common ground for the biophysical study of physiological phenomena that depend on transmembrane transport.

CA1 pyramidal cells have diverse biophysical properties, affected by development, experience, and aging

Neuron types (e.g., pyramidal cells) within one area of the brain are often considered homogeneous, despite variability in their biophysical properties. Here we review literature demonstrating variability in the electrical activity of CA1 hippocampal pyramidal cells (PCs), including responses to somatic current injection, synaptic stimulation, and spontaneous network-related activity. In addition, we describe how responses of CA1 PCs vary with development, experience, and aging, and some of the underlying ionic currents responsible. Finally, we suggest directions that may be the most impactful in expanding this knowledge, including the use of text and data mining to systematically study cellular heterogeneity in more depth; dynamical systems theory to understand and potentially classify neuron firing patterns; and mathematical modeling to study the interaction between cellular properties and network output. Our goals are to provide a synthesis of the literature for experimentalists studying CA1 PCs, to give theorists an idea of the rich diversity of behaviors models may need to reproduce to accurately represent these cells, and to provide suggestions for future research.

Neuronal Network Models for Sensory Discrimination

Previous modeling studies have demonstrated that lateral inhibition contributes to enhanced precision in sensory networks. That is, inhibitory connections reduce the spread of activity and repress neighboring cells, increasing the reliability of a sensory response. However, much less is understood about how connections that spread activity might contribute to the processing of sensory stimuli in the context of a sensory discrimination task. In this work, we examine the role of excitatory connections and gap junctions in network dynamics and some contributions to sensory discrimination.

Modeling the Influence of Ion Channels on Neuron Dynamics in Drosophila

Voltage gated ion channels play a major role in determining a neuron's firing behavior, resulting in the specific processing of synaptic input patterns. Drosophila and other invertebrates provide valuable model systems for investigating ion channel kinetics and their impact on firing properties. Despite the increasing importance of Drosophila as a model system, few computational models of its ion channel kinetics have been developed. In this study, experimentally observed biophysical properties of voltage gated ion channels from the fruitfly Drosophila melanogaster are used to develop a minimal, conductance based neuron model. We investigate the impact of the densities of these channels on the excitability of the model neuron. Changing the channel densities reproduces different in situ observed firing patterns and induces a switch from integrator to resonator properties. Further, we analyze the preference to input frequency and how it depends on the channel densities and the resulting bifurcation type the system undergoes. An extension to a three dimensional model demonstrates that the inactivation kinetics of the sodium channels play an important role, allowing for firing patterns with a delayed first spike and subsequent high frequency firing as often observed in invertebrates, without altering the kinetics of the delayed rectifier current.

Distal spike initiation zone location estimation by morphological simulation of ionic current filtering demonstrated in a novel model of an identified Drosophila motoneuron

Studying ion channel currents generated distally from the recording site is difficult because of artifacts caused by poor space clamp and membrane filtering. A computational model can quantify artifact parameters for correction by simulating the currents only if their exact anatomical location is known. We propose that the same artifacts that confound current recordings can help pinpoint the source of those currents by providing a signature of the neuron’s morphology. This method can improve the recording quality of currents initiated at the spike initiation zone (SIZ) that are often distal to the soma in invertebrate neurons. Drosophila being a valuable tool for characterizing ion currents, we estimated the SIZ location and quantified artifacts in an identified motoneuron, aCC/MN1-Ib, by constructing a novel multicompartmental model. Initial simulation of the measured biophysical channel properties in an isopotential Hodgkin-Huxley type neuron model partially replicated firing characteristics. Adding a second distal compartment, which contained spike-generating Na⁺ and K⁺ currents, was sufficient to simulate aCC’s in vivo activity signature. Matching this signature using a reconstructed morphology predicted that the SIZ is on aCC’s primary axon, 70 μm after the most distal dendritic branching point. From SIZ to soma, we observed and quantified selective morphological filtering of fast activating currents. Non-inactivating K⁺ currents are filtered ∼3 times less and despite their large magnitude at the soma they could be as distal as Na⁺ currents. The peak of transient component (NaT) of the voltage-activated Na⁺ current is also filtered more than the magnitude of slower persistent component (NaP), which can contribute to seizures. The corrected NaP/NaT ratio explains the previously observed discrepancy when the same channel is expressed in different cells. In summary, we used an in vivo signature to estimate ion channel location and recording artifacts, which can be applied to other neurons.

The study of ion channels is essential both for understanding normal brain function and for finding drug targets to treat neurological disease. Traditional experimental techniques remain challenging for recording ion channel currents accurately because of their locations in the neuron. Computer modeling of the three dimensional structure of neurons can provide a correction estimate for the measurement error introduced by neuronal membranes. To achieve this, we developed a modeling approach to localize, and correct for, distant ion channels. We demonstrated this approach by constructing novel computer models of an identified insect motor neuron, which provides a powerful model for studying the central nervous system. Through the study of electrical activity and genetic manipulations, it has been found that the persistent sodium current contributes to seizure. By modeling three dimensional structure, we were able to predict the location of these currents in the neuron, which were more distal than expected. Localizing sodium channels allowed us to predict their properties at origin, which favorably matched isolated recordings of these channels in more compact cells. This result is important in validating the use of heterologous compact cells to study insect sodium channels, and also demonstrates the usefulness of the presented modeling approach for studying channel physiology more generally.

Mathematical analysis of depolarization block mediated by slow inactivation of fast sodium channels in midbrain dopamine neurons

Dopamine neurons in freely moving rats often fire behaviorally relevant high-frequency bursts, but depolarization block limits the maximum steady firing rate of dopamine neurons in vitro to ∼10 Hz. Using a reduced model that faithfully reproduces the sodium current measured in these neurons, we show that adding an additional slow component of sodium channel inactivation, recently observed in these neurons, qualitatively changes in two different ways how the model enters into depolarization block. First, the slow time course of inactivation allows multiple spikes to be elicited during a strong depolarization prior to entry into depolarization block. Second, depolarization block occurs near or below the spike threshold, which ranges from -45 to -30 mV in vitro, because the additional slow component of inactivation negates the sodium window current. In the absence of the additional slow component of inactivation, this window current produces an N-shaped steady-state current-voltage (I-V) curve that prevents depolarization block in the experimentally observed voltage range near -40 mV. The time constant of recovery from slow inactivation during the interspike interval limits the maximum steady firing rate observed prior to entry into depolarization block. These qualitative features of the entry into depolarization block can be reversed experimentally by replacing the native sodium conductance with a virtual conductance lacking the slow component of inactivation. We show that the activation of NMDA and AMPA receptors can affect bursting and depolarization block in different ways, depending upon their relative contributions to depolarization versus to the total linear/nonlinear conductance.

Correlations in ion channel expression emerge from homeostatic tuning rules

Experimental observations reveal that the expression levels of different ion channels vary across neurons of a defined type, even when these neurons exhibit stereotyped electrical properties. However, there are robust correlations between different ion channel expression levels, although the mechanisms that determine these correlations are unknown. Using generic model neurons, we show that correlated conductance expression can emerge from simple homeostatic control mechanisms that couple expression rates of individual conductances to cellular readouts of activity. The correlations depend on the relative rates of expression of different conductances. Thus, variability is consistent with homeostatic regulation and the structure of this variability reveals quantitative relations between regulation dynamics of different conductances. Furthermore, we show that homeostatic regulation is remarkably insensitive to the details that couple the regulation of a given conductance to overall neuronal activity because of degeneracy in the function of multiple conductances and can be robust to "antihomeostatic" regulation of a subset of conductances expressed in a cell.