Interactions of persistent sodium and calcium-activated nonspecific cationic currents yield dynamically distinct bursting regimes in a model of respiratory neurons

The preBötzinger complex (preBötC) is a heterogeneous neuronal network within the mammalian brainstem that has been experimentally found to generate robust, synchronous bursts that drive the inspiratory phase of the respiratory rhythm. The persistent sodium (NaP) current is observed in every preBötC neuron, and significant modeling effort has characterized its contribution to square-wave bursting in the preBötC. Recent experimental work demonstrated that neurons within the preBötC are endowed with a calcium-activated nonspecific cationic (CAN) current that is activated by a signaling cascade initiated by glutamate. In a preBötC model, the CAN current was shown to promote robust bursts that experience depolarization block (DB bursts). We consider a self-coupled model neuron, which we represent as a single compartment based on our experimental finding of electrotonic compactness, under variation of g _NaP, the conductance of the NaP current, and g _CAN, the conductance of the CAN current. Varying these two conductances yields a spectrum of activity patterns, including quiescence, tonic activity, square-wave bursting, DB bursting, and a novel mixture of square-wave and DB bursts, which match well with activity that we observe in experimental preparations. We elucidate the mechanisms underlying these dynamics, as well as the transitions between these regimes and the occurrence of bistability, by applying the mathematical tools of bifurcation analysis and slow-fast decomposition. Based on the prevalence of NaP and CAN currents, we expect that the generalizable framework for modeling their interactions that we present may be relevant to the rhythmicity of other brain areas beyond the preBötC as well.     

The influence of sodium and potassium dynamics on excitability,seizures, and the stability of persistent states: I. Single neuron dynamics

In these companion papers, we study how the interrelated dynamics of sodium and potassium affect the excitability of neurons, the occurrence of seizures, and the stability of persistent states of activity. In this first paper, we construct a mathematical model consisting of a single conductance-based neuron together with intra- and extracellular ion concentration dynamics. We formulate a reduction of this model that permits a detailed bifurcation analysis, and show that the reduced model is a reasonable approximation of the full model. We find that competition between intrinsic neuronal currents, sodium-potassium pumps, glia, and diffusion can produce very slow and large-amplitude oscillations in ion concentrations similar to what is seen physiologically in seizures. Using the reduced model, we identify the dynamical mechanisms that give rise to these phenomena. These models reveal several experimentally testable predictions. Our work emphasizes the critical role of ion concentration homeostasis in the proper functioning of neurons, and points to important fundamental processes that may underlie pathological states such as epilepsy.

Convergent neuromodulation onto a network neuron can have divergent effects at the network level

Different neuromodulators often target the same ion channel. When such modulators act on different neuron types, this convergent action can enable a rhythmic network to produce distinct outputs. Less clear are the functional consequences when two neuromodulators influence the same ion channel in the same neuron. We examine the consequences of this seeming redundancy using a mathematical model of the crab gastric mill (chewing) network. This network is activated in vitro by the projection neuron MCN1, which elicits a half-center bursting oscillation between the reciprocally-inhibitory neurons LG and Int1. We focus on two neuropeptides which modulate this network, including a MCN1 neurotransmitter and the hormone crustacean cardioactive peptide (CCAP). Both activate the same voltage-gated current (I _MI ) in the LG neuron. However, I _MI-MCN1 , resulting from MCN1 released neuropeptide, has phasic dynamics in its maximal conductance due to LG presynaptic inhibition of MCN1, while I _MI-CCAP retains the same maximal conductance in both phases of the gastric mill rhythm. Separation of time scales allows us to produce a 2D model from which phase plane analysis shows that, as in the biological system, I _MI-MCN1 and I _MI-CCAP primarily influence the durations of opposing phases of this rhythm. Furthermore, I _MI-MCN1 influences the rhythmic output in a manner similar to the Int1-to-LG synapse, whereas I _MI-CCAP has an influence similar to the LG-to-Int1 synapse. These results show that distinct neuromodulators which target the same voltage-gated ion channel in the same network neuron can nevertheless produce distinct effects at the network level, providing divergent neuromodulator actions on network activity.     

Modeling the effects of extracellular potassium on bursting properties in pre-Bötzinger complex neurons

There are many types of neurons that intrinsically generate rhythmic bursting activity, even when isolated, and these neurons underlie several specific motor behaviors. Rhythmic neurons that drive the inspiratory phase of respiration are located in the medullary pre-Bötzinger Complex (pre-BötC). However, it is not known if their rhythmic bursting is the result of intrinsic mechanisms or synaptic interactions. In many cases, for bursting to occur, the excitability of these neurons needs to be elevated. This excitation is provided in vitro (e.g. in slices), by increasing extracellular potassium concentration (K _out ) well beyond physiologic levels. Elevated K _out shifts the reversal potentials for all potassium currents including the potassium component of leakage to higher values. However, how an increase in K _out , and the resultant changes in potassium currents, induce bursting activity, have yet to be established. Moreover, it is not known if the endogenous bursting induced in vitro is representative of neural behavior in vivo. Our modeling study examines the interplay between K _out , excitability, and selected currents, as they relate to endogenous rhythmic bursting. Starting with a Hodgkin-Huxley formalization of a pre-BötC neuron, a potassium ion component was incorporated into the leakage current, and model behaviors were investigated at varying concentrations of K _out . Our simulations show that endogenous bursting activity, evoked in vitro by elevation of K _out , is the result of a specific relationship between the leakage and voltage-dependent, delayed rectifier potassium currents, which may not be observed at physiological levels of extracellular potassium.     

The influence of sodium and potassium dynamics on excitability,seizures, and the stability of persistent states: II. Network and glial dynamics

In these companion papers, we study how the interrelated dynamics of sodium and potassium affect the excitability of neurons, the occurrence of seizures, and the stability of persistent states of activity. We seek to study these dynamics with respect to the following compartments: neurons, glia, and extracellular space. We are particularly interested in the slower time-scale dynamics that determine overall excitability, and set the stage for transient episodes of persistent oscillations, working memory, or seizures. In this second of two companion papers, we present an ionic current network model composed of populations of Hodgkin–Huxley type excitatory and inhibitory neurons embedded within extracellular space and glia, in order to investigate the role of micro-environmental ionic dynamics on the stability of persistent activity. We show that these networks reproduce seizure-like activity if glial cells fail to maintain the proper micro-environmental conditions surrounding neurons, and produce several experimentally testable predictions. Our work suggests that the stability of persistent states to perturbation is set by glial activity, and that how the response to such perturbations decays or grows may be a critical factor in a variety of disparate transient phenomena such as working memory, burst firing in neonatal brain or spinal cord, up states, seizures, and cortical oscillations.

Rules of forming receptive fields of formal neurons during unsupervised learning processes

In the paper two kinds of unsupervised learning processes occurring in formal neurons are analysed. The relationship between these processes and the supervised ones is discussed too. In the first case of unsupervised learning processes the neuron is considered as a filter that passes signals most frequently occurring in the learning sequence {x[n]}. In the second case it is considered as a detector of rareness which, after a finite number of steps operates as a filter passing only signals which rarely occur in the learning sequence {x[n]}. These two approaches result in different types of receptive fields of formal neurons. On the basis of the results obtained, it is possible to advance a hypothesis on the role of neurons with various types of receptive fields in information processing by a complex neuronal network.     

Spike-time reliability of layered neural oscillator networks

We study the reliability of layered networks of coupled “type I” neural oscillators in response to fluctuating input signals. Reliability means that a signal elicits essentially identical responses upon repeated presentations, regardless of the network’s initial condition. We study reliability on two distinct scales: neuronal reliability, which concerns the repeatability of spike times of individual neurons embedded within a network, and pooled-response reliability, which concerns the repeatability of total synaptic outputs from a subpopulation of the neurons in a network. We find that neuronal reliability depends strongly both on the overall architecture of a network, such as whether it is arranged into one or two layers, and on the strengths of the synaptic connections. Specifically, for the type of single-neuron dynamics and coupling considered, single-layer networks are found to be very reliable, while two-layer networks lose their reliability with the introduction of even a small amount of feedback. As expected, pooled responses for large enough populations become more reliable, even when individual neurons are not. We also study the effects of noise on reliability, and find that noise that affects all neurons similarly has much greater impact on reliability than noise that affects each neuron differently. Qualitative explanations are proposed for the phenomena observed.

First return maps for the dynamics of synaptically coupled conditional bursters

The pre-Bötzinger complex (preBötc) in the mammalian brainstem has an important role in generating respiratory rhythms. An influential differential equation model for the activity of individual neurons in the preBötc yields transitions from quiescence to bursting to tonic spiking as a parameter is varied. Further, past work has established that bursting dynamics can arise from a pair of tonic model cells coupled with synaptic excitation. In this paper, we analytically derive one- and two-dimensional maps from the differential equations for a self-coupled neuron and a two-neuron network, respectively. Using a combination of analysis and simulations of these maps, we explore the possible forms of dynamics that the model networks can produce as well as which transitions between dynamic regimes are mathematically possible.     

Self-organization of day cycle and hierarchical associative memory in “live” neural network

The live neural network model is proposed on the basis of live neuron model and optimal learning rule. By means of numerical simulation the initial stages of neural network self-organization have been shown: (1) the formation of two activity forms, which are identified with sleep and awaking, and (2) the self-organization of hierarchical associative memory when feeding a receptor excitation to the neural network. The energetic profit of self-organization is demonstrated. The formation of neural ensembles, playing the role of generalized neurons, is obtained.     

Properties of Proton-Gated Ion Channels in Sensory Neurons of Rats

In experiments on the somata of sensory neurons isolated from the spinal and trigeminal ganglia of rats, we characterized three subclasses of proton-gated currents differing from each other in their kinetics of desensitization and characteristics of stationary desensitization (but not in the characteristics of stationary activation). A voltage clamp technique in the whole cell configuration and intracellular perfusion were used. Expression of the channels providing currents of each subclass depended on the soma diameter but not on anatomical localization of the neuron. Proton-gated channels of type I were characterized by mono- or biexponential kinetics of current desensitization with the duration of complete decay within a 1 to 15 sec range; the mean pH₅₀ of the curve of stationary desensitization was 7.21 ± 0.02. Channels of type II possessed mostly monoexponential desensitization kinetics with the duration of decay within a 1 to 3 sec range; their pH₅₀ of the stationary desensitization curve was 7.11 ± 0.02. Channels of type III showed mostly biexponential desensitization kinetics; the complete current decay lasted about 5 sec, while the mean pH₅₀ was about 6.78 ± 0.02. Channels of type I were typical of small neurons (soma diameter 10-20 m), while those of types II and III were found mostly in large cells (35-60 m).     

Dynamics of spontaneous activity in random networks with multiple neuron subtypes and synaptic noise

During slow-wave sleep, brain electrical activity is dominated by the slow (< 1 Hz) electroencephalogram (EEG) oscillations characterized by the periodic transitions between active (or Up) and silent (or Down) states in the membrane voltage of the cortical and thalamic neurons. Sleep slow oscillation is believed to play critical role in consolidation of recent memories. Past computational studies, based on the Hodgkin-Huxley type neuronal models, revealed possible intracellular and network mechanisms of the neuronal activity during sleep, however, they failed to explore the large-scale cortical network dynamics depending on collective behavior in the large populations of neurons. In this new study, we developed a novel class of reduced discrete time spiking neuron models for large-scale network simulations of wake and sleep dynamics. In addition to the spiking mechanism, the new model implemented nonlinearities capturing effects of the leak current, the Ca²⁺ dependent K⁺ current and the persistent Na⁺ current that were found to be critical for transitions between Up and Down states of the slow oscillation. We applied the new model to study large-scale two-dimensional cortical network activity during slow-wave sleep. Our study explained traveling wave dynamics and characteristic synchronization properties of transitions between Up and Down states of the slow oscillation as observed in vivo in recordings from cats. We further predict a critical role of synaptic noise and slow adaptive currents for spike sequence replay as found during sleep related memory consolidation.     

Impact of slow K<Superscript>+</Superscript> currents on spike generation can be described by an adaptive threshold model

A neuron that is stimulated by rectangular current injections initially responds with a high firing rate, followed by a decrease in the firing rate. This phenomenon is called spike-frequency adaptation and is usually mediated by slow K⁺ currents, such as the M-type K⁺ current (I _M ) or the Ca²⁺-activated K⁺ current (I _AHP ). It is not clear how the detailed biophysical mechanisms regulate spike generation in a cortical neuron. In this study, we investigated the impact of slow K⁺ currents on spike generation mechanism by reducing a detailed conductance-based neuron model. We showed that the detailed model can be reduced to a multi-timescale adaptive threshold model, and derived the formulae that describe the relationship between slow K⁺ current parameters and reduced model parameters. Our analysis of the reduced model suggests that slow K⁺ currents have a differential effect on the noise tolerance in neural coding.     

On some properties of stochastic information processes in neurons and neuron populations

The information in the nervous spike trains and its processing by neural units are discussed. In these problems, our attention is focused on the stochastic properties of neurons and neuron populations. There are three subjects in this paper, which are the spontaneous type neuron, the forced type neuron and the reciprocal inhibitory pairs.

1. The spontaneous type neuron produces spikes without excitatory inputs. The mathematical model has the following assumptions. The neuron potential (NP) has the fluctuation and obeys the Ornstein-Uhlenbeck process, because the N P is not so perfectly random as that of the Wiener process but has an attraction to the rest value. The threshold varies exponentially and the NP has the constant lower limit. When the NP reaches the threshold, the neuron fires and the NP is reset to a certain position. After a firing, an absolute refractory period exists. In discussing the stochastic properties of neurons, the transition probability density function and the first passage time density function are the important quantities, which are governed by the Kolmogorov's equations. Although they can be set up easily, we can rarely obtain the analytical solutions in time domain. Moreover, they cover only simple properties. Hence the numerical analysis is performed and a good deal of fair results are obtained and discussed.
2. The forced type neuron has input pulse trains which are assumed to be based on the Poisson process. Other assumptions and methods are almost the same as above except the diffusion approximation of the stochastic process. In this case, we encounter the inhomogeneous process due to the pulse-frequency-modulation, whose first passage time density reveals the multimodal distribution. The numerical analysis is also tried, and the output spike interval density is further discussed in the case of the periodic modulation.
3. Two types of reciprocal inhibitory pairs are discussed. The first type has two excitatory driving inputs which are mutually independent. The second type has one common excitatory input but it advances in two ways, one of which has a time lag. The neuron dynamics is the same as that of the forced type neuron and each neuron has an identical structure. The inputs are assumed to be based on the Poisson process and the inhibition occurs when the companion neuron fires. In this case, the equations of the probability density functions are not obtained. Hence the computer simulation is tried and it is observed that the stochastic rhythm emerges in spite of the temporally homogeneous inputs. Furthermore, the case of inhomogeneous inputs is discussed.

     

The mechanism of mental scanning in foveal vision

In order to perceive a visual pattern which includes several elemental pictures, the perceiver must allot his cognitive resources to suitably selected parts of the pattern and scan them in sequence. Even when the visual field is small and eye-movement is not required, such scanning is found. We called it mental scanning and performed psychological experiments to investigate the mechanism. The tasks were to discern whether the elemental pictures in a pattern are all the same (SP) or not (DP). The per cents correct of the task were measured for various exposure durations. We defined the threshold as the exposure duration at which 75% correct answers were obtained. Our main findings are as follows. The threshold for SP is proportional to the number of picture elements, while the threshold for DP is constant. It appears that two modes of mental scanning exist. One is serial processing for SP, and the other is parallel processing for DP. We proposed a two-layered neural network model having the following characteristics. 1) Information is transmitted as two types of signals through two separate channels; one is the transient signals to the Y layer and the other is the sustained signals slowly conducted to the X layer. 2) Interactions among neurons in the Y layer are lateral inhibitory, while those in the X layer are self-excitatory and lateralinhibitory. 3) Every neuron in the Y layer sends inhibitory signals to every neuron in the X layer except one with the same receptive field. Under these conditions, the dynamics of neurons in the X layer is represented by a set of certain equations. From phase plane analysis and numerical integration, the model appears to have an ability to account for various experimental results.     

Synchronous Clusters in a Noisy Inhibitory Neural Network

We study the stability and information encoding capacity of synchronized states in a neuronal network model that represents part of thalamic circuitry. Our model neurons have a Hodgkin-Huxley-type low-threshold calcium channel, display postinhibitory rebound, and are connected via GABAergic inhibitory synapses.We find that there is a threshold in synaptic strength, _c, below which there are no stable spiking network states. Above threshold the stable spiking state is a cluster state, where different groups of neurons fire consecutively, and each neuron fires with the same cluster each time. Weak noise destabilizes this state, but stronger noise drives the system into a different, self-organized, stochastically synchronized state. Neuronal firing is still organized in clusters, but individual neurons can hop from cluster to cluster. Noise can actually induce and sustain such a state below the threshold of synaptic strength. We do find a qualitative difference in the firing patterns between small (10 neurons) and large (1000 neurons) networks.We determine the information content of the spike trains in terms of two separate contributions: the spike-time jitter around cluster firing times, and the hopping from cluster to cluster. We quantify the information loss due to temporally correlated interspike intervals. Recent experiments on the locust olfactory system and striatal neurons suggest that the nervous system may actually use these two channels to encode separate and unique information.     

Physiological,Morphological and Neurochemical Characterization of Neurons Modulated by Movement

The role of individual neurons and their function in neuronal circuits is fundamental to understanding the neuronal mechanisms of sensory and motor functions. Most investigations of sensorimotor mechanisms rely on either examination of neurons while an animal is static^1,2 or record extracellular neuronal activity during a movement.^3,4 While these studies have provided the fundamental background for sensorimotor function, they either do not evaluate functional information which occurs during a movement or are limited in their ability to fully characterize the anatomy, physiology and neurochemical phenotype of the neuron. A technique is shown here which allows extensive characterization of individual neurons during an in vivo movement. This technique can be used not only to study primary afferent neurons but also to characterize motoneurons and sensorimotor interneurons. Initially the response of a single neuron is recorded using electrophysiological methods during various movements of the mandible followed by determination of the receptive field for the neuron. A neuronal tracer is then intracellularly injected into the neuron and the brain is processed so that the neuron can be visualized with light, electron or confocal microscopy (Fig. 1). The detailed morphology of the characterized neuron is then reconstructed so that neuronal morphology can be correlated with the physiological response of the neuron (Figs. 2,3). In this communication important key details and tips for successful implementation of this technique are provided. Valuable additional information can be determined for the neuron under study by combining this method with other techniques. Retrograde neuronal labeling can be used to determine neurons with which the labeled neuron synapses; thus allowing detailed determination of neuronal circuitry. Immunocytochemistry can be combined with this method to examine neurotransmitters within the labeled neuron and to determine the chemical phenotypes of neurons with which the labeled neuron synapses. The labeled neuron can also be processed for electron microscopy to determine the ultrastructural features and microcircuitry of the labeled neuron. Overall this technique is a powerful method to thoroughly characterize neurons during in vivo movement thus allowing substantial insight into the role of the neuron in sensorimotor function.Download video file.^{(43M, mov)}     

Bistability,switches and working memory in a two-neuron inhibitory-feedback model

It was reported earlier that an inhibitory-feedback network inspired by neostriatal circuitry may exhibit a bistable character and spontaneous switching phenomenon within the neuronal activity. In the presence of noise and external excitation, a few local neurons switch on and generate streams of impulses while other neurons remain quiescent. In time, the existing on neurons spontaneously switch off and other neurons switch on. In this paper we examine the nature of the bistability and switching phenomenon using a simple model consisting of two mutually inhibitory neurons. For nonspiking neuron model, described by a system of nonlinear differential equations, we present a simple bifurcation analysis, which follows the birth and annihilation of two stable fixed points when model parameters are varied. We show that both nonspiking and spiking models may have two stable states, but only spiking neurons exhibit switching. The mechanism of switching for model spiking neurons, described by an equivalent RC circuit with a number of currents, is analyzed using computer simulations. It is shown that switching can be described by a two-state Markov chain with one parameter, which depends on the set of model physiological parameters, such as duration of afterhyperpolarization (AHP), maximum and the time duration of inhibitory post-synaptic potentials (IPSP's) and amplitude of the neuron noise input. On and off states of the model can be rapidly changed by localized excitatory input and the network then sustains the pattern of on and off states. That is, such a network can be used as a programmable memory device. Our hypothesis is that biological neural networks exhibit switches in their evolution to low energy states and switches are essential for the load and readout of the temporary and long term memory.     

Adaptation in Helix pomatia neurons

• 1.1. Spike frequency adaptation has been studied on neurons of Helix pomatia subesophageal ganglia and interpreted by means of a behavioural model describing the phenomenon in neurons either silent or autorhythmic at rest.
• 2.2. At low stimulating currents the initial discharge frequency F(0) is linearly related to the current strength G.
• 3.3. In the linearity range F(0)/G each neuron was characterized by means of four model parameters: the proportionality constant between F(0) and G, the decay constant of the frequency, the inhibitory current from a single nerve impulse and the decay time constant of the inhibitory current.
• 4.4. The four parameters varied in different cells with a range of 0.18–4.98 Hz/nA, 1.02–3.85 sec, 0.05–0.95 nA and 1.74–22.33 see, respectively.
• 5.5. Experimental results have been analyzed considering inhibitory current, electrogenie sodium pump and other proposed adaptation parameters.

     

Modeling brain dynamics using computational neurogenetic approach

The paper introduces a novel computational approach to brain dynamics modeling that integrates dynamic gene–protein regulatory networks with a neural network model. Interaction of genes and proteins in neurons affects the dynamics of the whole neural network. Through tuning the gene–protein interaction network and the initial gene/protein expression values, different states of the neural network dynamics can be achieved. A generic computational neurogenetic model is introduced that implements this approach. It is illustrated by means of a simple neurogenetic model of a spiking neural network of the generation of local field potential. Our approach allows for investigation of how deleted or mutated genes can alter the dynamics of a model neural network. We conclude with the proposal how to extend this approach to model cognitive neurodynamics.

Quantal glutamate release is essential for reliable neuronal encodings in cerebral networks

Background

The neurons and synapses work coordinately to program the brain codes of controlling cognition and behaviors. Spike patterns at the presynaptic neurons regulate synaptic transmission. The quantitative regulations of synapse dynamics in spike encoding at the postsynaptic neurons remain unclear.

Methodology/Principal Findings

With dual whole-cell recordings at synapse-paired cells in mouse cortical slices, we have investigated the regulation of synapse dynamics to neuronal spike encoding at cerebral circuits assembled by pyramidal neurons and GABAergic ones. Our studies at unitary synapses show that postsynaptic responses are constant over time, such as glutamate receptor-channel currents at GABAergic neurons and glutamate transport currents at astrocytes, indicating quantal glutamate release. In terms of its physiological impact, our results demonstrate that the signals integrated from quantal glutamatergic synapses drive spike encoding at GABAergic neurons reliably, which in turn precisely set spike encoding at pyramidal neurons through feedback inhibition.

Conclusion/Significance

Our studies provide the evidences for the quantal glutamate release to drive the spike encodings precisely in cortical circuits, which may be essential for programming the reliable codes in the brain to manage well-organized behaviors.