Effects of ion channel noise on neural circuits: an application to the respiratory pattern generator to investigate breathing variability

Simple model to incorporate statistical noise based on a modified hodgkin-huxley approach for external electrical field driven neural responses

Noise activity is known to affect neural networks, enhance the system response to weak external signals, and lead to stochastic resonance phenomenon that can effectively amplify signals in nonlinear systems. In most treatments, channel noise has been modeled based on multi-state Markov descriptions or the use stochastic differential equation models. Here we probe a computationally simple approach based on a minor modification of the traditional Hodgkin-Huxley approach to embed noise in neural response. Results obtained from numerous simulations with different excitation frequencies and noise amplitudes for the action potential firing show very good agreement with output obtained from well-established models. Furthermore, results from the Mann-Whitney U Test reveal a statistically insignificant difference. The distribution of the time interval between successive potential spikes obtained from this simple approach compared very well with the results of complicated Fox and Lu type methods at much reduced computational cost. This present method could also possibly be applied to the analysis of spatial variations and/or differences in characteristics of random incident electromagnetic signals.

Temporal variations in the pattern of breathing: techniques, sources, and applications to translational sciences

The breathing process possesses a complex variability caused in part by the respiratory central pattern generator in the brainstem; however, it also arises from chemical and mechanical feedback control loops, network reorganization and network sharing with nonrespiratory motor acts, as well as inputs from cortical and subcortical systems. The notion that respiratory fluctuations contain hidden information has prompted scientists to decipher respiratory signals to better understand the fundamental mechanisms of respiratory pattern generation, interactions with emotion, influences on the cortical neuronal networks associated with cognition, and changes in variability in healthy and disease-carrying individuals. Respiration can be used to express and control emotion. Furthermore, respiration appears to organize brain-wide network oscillations via cross-frequency coupling, optimizing cognitive performance. With the aid of information theory-based techniques and machine learning, the hidden information can be translated into a form usable in clinical practice for diagnosis, emotion recognition, and mental conditioning.

Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics

Fox and Lu introduced a Langevin framework for discrete-time stochastic models of randomly gated ion channels such as the Hodgkin-Huxley (HH) system. They derived a Fokker-Planck equation with state-dependent diffusion tensor D and suggested a Langevin formulation with noise coefficient matrix S such that SS⊤=D. Subsequently, several authors introduced a variety of Langevin equations for the HH system. In this article, we present a natural 14-dimensional dynamics for the HH system in which each directed edge in the ion channel state transition graph acts as an independent noise source, leading to a 14 × 28 noise coefficient matrix S. We show that (1) the corresponding 14D system of ordinary differential equations is consistent with the classical 4D representation of the HH system; (2) the 14D representation leads to a noise coefficient matrix S that can be obtained cheaply on each time step, without requiring a matrix decomposition; (3) sample trajectories of the 14D representation are pathwise equivalent to trajectories of Fox and Lu's system, as well as trajectories of several existing Langevin models; (4) our 14D representation (and those equivalent to it) gives the most accurate interspike interval distribution, not only with respect to moments but under both the L1 and L∞ metric-space norms; and (5) the 14D representation gives an approximation to exact Markov chain simulations that are as fast and as efficient as all equivalent models. Our approach goes beyond existing models, in that it supports a stochastic shielding decomposition that dramatically simplifies S with minimal loss of accuracy under both voltage- and current-clamp conditions.

Volumetric mapping of the functional neuroanatomy of the respiratory network in the perfused brainstem preparation of rats

KEY POINTS

The functional neuroanatomy of the mammalian respiratory network is far from being understood since experimental tools that measure neural activity across this brainstem-wide circuit are lacking. Here, we use silicon multi-electrode arrays to record respiratory local field potentials (rLFPs) from 196-364 electrode sites within 8-10 mm³ of brainstem tissue in single arterially perfused brainstem preparations with respect to the ongoing respiratory motor pattern of inspiration (I), post-inspiration (PI) and late-expiration (E2). rLFPs peaked specifically at the three respiratory phase transitions, E2-I, I-PI and PI-E2. We show, for the first time, that only the I-PI transition engages a brainstem-wide network, and that rLFPs during the PI-E2 transition identify a hitherto unknown role for the dorsal respiratory group. Volumetric mapping of pontomedullary rLFPs in single preparations could become a reliable tool for assessing the functional neuroanatomy of the respiratory network in health and disease.

ABSTRACT

While it is widely accepted that inspiratory rhythm generation depends on the pre-Bötzinger complex, the functional neuroanatomy of the neural circuits that generate expiration is debated. We hypothesized that the compartmental organization of the brainstem respiratory network is sufficient to generate macroscopic local field potentials (LFPs), and if so, respiratory (r) LFPs could be used to map the functional neuroanatomy of the respiratory network. We developed an approach using silicon multi-electrode arrays to record spontaneous LFPs from hundreds of electrode sites in a volume of brainstem tissue while monitoring the respiratory motor pattern on phrenic and vagal nerves in the perfused brainstem preparation. Our results revealed the expression of rLFPs across the pontomedullary brainstem. rLFPs occurred specifically at the three transitions between respiratory phases: (1) from late expiration (E2) to inspiration (I), (2) from I to post-inspiration (PI), and (3) from PI to E2. Thus, respiratory network activity was maximal at respiratory phase transitions. Spatially, the E2-I, and PI-E2 transitions were anatomically localized to the ventral and dorsal respiratory groups, respectively. In contrast, our data show, for the first time, that the generation of controlled expiration during the post-inspiratory phase engages a distributed neuronal population within ventral, dorsal and pontine network compartments. A group-wise independent component analysis demonstrated that all preparations exhibited rLFPs with a similar temporal structure and thus share a similar functional neuroanatomy. Thus, volumetric mapping of rLFPs could allow for the physiological assessment of global respiratory network organization in health and disease.

Experimental Validation of a Closed-Loop Respiratory Control Model using Dynamic Clamp

We have previously introduced a model for closed-loop respiratory control incorporating an explicit conductance-based model of bursting pacemaker cells driven by hypoxia sensitive chemosensory feedback. Numerical solution of the model equations revealed two qualitatively distinct asymptotically stable dynamical behaviors: one analogous to regular breathing (eupnea), and a second analogous to pathologically rapid, shallow breathing (tachypnea). As an experimental test of this model, we created a hybrid in vitrolin silico circuit. We used Real Time eXperimental Interface (RTXI) dynamic clamp to incorporate a living pacemaker cell recorded in vitro into a numerical simulation of the closed-loop control model in real time. Here we show that the hybrid circuit can sustain the same bistable behavior as the purely computational model, and we assess the ability of the hybrid circuit to recover from simulated bouts of transient hypoxia.

Effects of persistent sodium current blockade in respiratory circuits depend on the pharmacological mechanism of action and network dynamics

The mechanism(s) of action of most commonly used pharmacological blockers of voltage-gated ion channels are well understood; however, this knowledge is rarely considered when interpreting experimental data. Effects of blockade are often assumed to be equivalent, regardless of the mechanism of the blocker involved. Using computer simulations, we demonstrate that this assumption may not always be correct. We simulate the blockade of a persistent sodium current (I_NaP), proposed to underlie rhythm generation in pre-Bötzinger complex (pre-BötC) respiratory neurons, via two distinct pharmacological mechanisms: (1) pore obstruction mediated by tetrodotoxin and (2) altered inactivation dynamics mediated by riluzole. The reported effects of experimental application of tetrodotoxin and riluzole in respiratory circuits are diverse and seemingly contradictory and have led to considerable debate within the field as to the specific role of I_NaP in respiratory circuits. The results of our simulations match a wide array of experimental data spanning from the level of isolated pre-BötC neurons to the level of the intact respiratory network and also generate a series of experimentally testable predictions. Specifically, in this study we: (1) provide a mechanistic explanation for seemingly contradictory experimental results from in vitro studies of I_NaP block, (2) show that the effects of I_NaP block in in vitro preparations are not necessarily equivalent to those in more intact preparations, (3) demonstrate and explain why riluzole application may fail to effectively block I_NaP in the intact respiratory network, and (4) derive the prediction that effective block of I_NaP by low concentration tetrodotoxin will stop respiratory rhythm generation in the intact respiratory network. These simulations support a critical role for I_NaP in respiratory rhythmogenesis in vivo and illustrate the importance of considering mechanism when interpreting and simulating data relating to pharmacological blockade.

The application of pharmacological agents that affect transmembrane ionic currents in neurons is a commonly used experimental technique. A simplistic interpretation of experiments involving these agents suggests that antagonist application removes the impacted current and that subsequently observed changes in activity are attributable to the loss of that current’s effects. The more complex reality, however, is that different drugs may have distinct mechanisms of action, some corresponding not to a removal of a current but rather to a changing of its properties. We use computational modeling to explore the implications of the distinct mechanisms associated with two drugs, riluzole and tetrodotoxin, that are often characterized as sodium channel blockers. Through this approach, we offer potential explanations for disparate findings observed in experiments on neural respiratory circuits and show that the experimental results are consistent with a key role for the persistent sodium current in respiratory rhythm generation.

Fluctuation Scaling of Neuronal Firing and Bursting in Spontaneously Active Brain Circuits

We employed high-density microelectrode arrays to investigate spontaneous firing patterns of neurons in brain circuits of the primary somatosensory cortex (S1) in mice. We recorded from over 150 neurons for 10min in each of eight different experiments, identified their location in S1, sorted their action potentials (spikes), and computed their power spectra and inter-spike interval (ISI) statistics. Of all persistently active neurons, 92% fired with a single dominant frequency - regularly firing neurons (RNs) - from 1 to 8Hz while 8% fired in burst with two dominant frequencies - bursting neurons (BNs) - corresponding to the inter-burst (2-6Hz) and intra-burst intervals (20-160Hz). RNs were predominantly located in layers 2/3 and 5/6 while BNs localized to layers 4 and 5. Across neurons, the standard deviation of ISI was a power law of its mean, a property known as fluctuation scaling, with a power law exponent of 1 for RNs and 1.25 for BNs. The power law implies that firing and bursting patterns are scale invariant: the firing pattern of a given RN or BN resembles that of another RN or BN, respectively, after a time contraction or dilation. An explanation for this scale invariance is discussed in the context of previous computational studies as well as its potential role in information processing.

Stochastic shielding and edge importance for Markov chains with timescale separation

Nerve cells produce electrical impulses (“spikes”) through the coordinated opening and closing of ion channels. Markov processes with voltage-dependent transition rates capture the stochasticity of spike generation at the cost of complex, time-consuming simulations. Schmandt and Galán introduced a novel method, based on the stochastic shielding approximation, as a fast, accurate method for generating approximate sample paths with excellent first and second moment agreement to exact stochastic simulations. We previously analyzed the mathematical basis for the method’s remarkable accuracy, and showed that for models with a Gaussian noise approximation, the stationary variance of the occupancy at each vertex in the ion channel state graph could be written as a sum of distinct contributions from each edge in the graph. We extend this analysis to arbitrary discrete population models with first-order kinetics. The resulting decomposition allows us to rank the “importance” of each edge’s contribution to the variance of the current under stationary conditions. In most cases, transitions between open (conducting) and closed (non-conducting) states make the greatest contributions to the variance, but there are exceptions. In a 5-state model of the nicotinic acetylcholine receptor, at low agonist concentration, a pair of “hidden” transitions (between two closed states) makes a greater contribution to the variance than any of the open-closed transitions. We exhaustively investigate this “edge importance reversal” phenomenon in simplified 3-state models, and obtain an exact formula for the contribution of each edge to the variance of the open state. Two conditions contribute to reversals: the opening rate should be faster than all other rates in the system, and the closed state leading to the opening rate should be sparsely occupied. When edge importance reversal occurs, current fluctuations are dominated by a slow noise component arising from the hidden transitions.

Discrete state, continuous time Markov processes occur throughout cell biology, neuroscience, and ecology, representing the random dynamics of processes transitioning among multiple locations or states. Complexity reduction for such models aims to capture the essential dynamics and stochastic properties via a simpler representation, with minimal loss of accuracy. Classical approaches, such as aggregation of nodes and elimination of fast variables, lead to reduced models that are no longer Markovian. Stochastic shielding provides an alternative approach by simplifying the description of the noise driving the process, while preserving the Markov property, by removing from the model those fluctuations that are not directly observable. We previously applied the stochastic shielding approximation to several Markov processes arising in neuroscience and processes on random graphs. Here we explore the range of validity of stochastic shielding for processes with nonuniform stationary probabilities and multiple timescales, including ion channels with “bursty” dynamics. We show that stochastic shielding is robust to the introduction of timescale separation, for a class of simple networks, but it can break down for more complex systems with three distinct timescales. We also show that our related edge importance measure remains valid for arbitrary networks regardless of multiple timescales.

REM sleep respiratory behaviours mental content in narcoleptic lucid dreamers

Breathing is irregular during rapid eye-movement (REM) sleep, whereas it is stable during non-REM sleep. Why this is so remains a mystery. We propose that irregular breathing has a cortical origin and reflects the mental content of dreams, which often accompany REM sleep. We tested 21 patients with narcolepsy who had the exceptional ability to lucid dream in REM sleep, a condition in which one is conscious of dreaming during the dream and can signal lucidity with an ocular code. Sleep and respiration were monitored during multiple naps. Participants were instructed to modify their dream scenario so that it involved vocalizations or an apnoea, -two behaviours that require a cortical control of ventilation when executed during wakefulness. Most participants (86%) were able to signal lucidity in at least one nap. In 50% of the lucid naps, we found a clear congruence between the dream report (e.g., diving under water) and the observed respiratory behaviour (e.g., central apnoea) and, in several cases, a preparatory breath before the respiratory behaviour. This suggests that the cortico-subcortical networks involved in voluntary respiratory movements are preserved during REM sleep and that breathing irregularities during this stage have a cortical/subcortical origin that reflects dream content.

Anti-correlated cortical networks arise from spontaneous neuronal dynamics at slow timescales

In the highly interconnected architectures of the cerebral cortex, recurrent intracortical loops disproportionately outnumber thalamo-cortical inputs. These networks are also capable of generating neuronal activity without feedforward sensory drive. It is unknown, however, what spatiotemporal patterns may be solely attributed to intrinsic connections of the local cortical network. Using high-density microelectrode arrays, here we show that in the isolated, primary somatosensory cortex of mice, neuronal firing fluctuates on timescales from milliseconds to tens of seconds. Slower firing fluctuations reveal two spatially distinct neuronal ensembles, which correspond to superficial and deeper layers. These ensembles are anti-correlated: when one fires more, the other fires less and vice versa. This interplay is clearest at timescales of several seconds and is therefore consistent with shifts between active sensing and anticipatory behavioral states in mice.

Kölliker-Fuse nuclei regulate respiratory rhythm variability via a gain-control mechanism

Respiration varies from breath to breath. On the millisecond timescale of spiking, neuronal circuits exhibit variability due to the stochastic properties of ion channels and synapses. Does this fast, microscopic source of variability contribute to the slower, macroscopic variability of the respiratory period? To address this question, we modeled a stochastic oscillator with forcing; then, we tested its predictions experimentally for the respiratory rhythm generated by the in situ perfused preparation during vagal nerve stimulation (VNS). Our simulations identified a relationship among the gain of the input, entrainment strength, and rhythm variability. Specifically, at high gain, the periodic input entrained the oscillator and reduced variability, whereas at low gain, the noise interacted with the input, causing events known as "phase slips", which increased variability on a slow timescale. Experimentally, the in situ preparation behaved like the low-gain model: VNS entrained respiration but exhibited phase slips that increased rhythm variability. Next, we used bilateral muscimol microinjections in discrete respiratory compartments to identify areas involved in VNS gain control. Suppression of activity in the nucleus tractus solitarii occluded both entrainment and amplification of rhythm variability by VNS, confirming that these effects were due to the activation of the Hering-Breuer reflex. Suppressing activity of the Kölliker-Fuse nuclei (KFn) enhanced entrainment and reduced rhythm variability during VNS, consistent with the predictions of the high-gain model. Together, the model and experiments suggest that the KFn regulates respiratory rhythm variability via a gain control mechanism.