Dopamine depletion weakens direct pathway modulation of SNr neurons.

Neurons in the substantia nigra reticulata (SNr) transmit information about basal ganglia output to dozens of brain regions in thalamocortical and brainstem motor networks. Activity of SNr neurons is regulated by convergent input from upstream basal ganglia nuclei, including GABAergic inputs from the striatum and the external globus pallidus (GPe). GABAergic inputs from the striatum convey information from the direct pathway, while GABAergic inputs from the GPe convey information from the indirect pathway. Chronic loss of dopamine, as occurs in Parkinson's disease, disrupts the balance of direct and indirect pathway neurons at the level of the striatum, but the question of how dopamine loss affects information propagation along these pathways outside of the striatum is less well understood. Using a combination of in vivo and slice electrophysiology, we find that dopamine depletion selectively weakens the direct pathway's influence over neural activity in the SNr due to changes in the decay kinetics of GABA-mediated synaptic currents. GABAergic signaling from GPe neurons in the indirect pathway was not affected, resulting in an inversion of the normal balance of inhibitory control over basal ganglia output through the SNr. These results highlight the contribution of cellular mechanisms outside of the striatum that impact the responses of basal ganglia output neurons to the direct and indirect pathways in disease.

Rethinking the external globus pallidus and information flow in cortico-basal ganglia-thalamic circuits.

For decades, the external globus pallidus (GPe) has been viewed as a passive way-station in the indirect pathway of the cortico-basal ganglia-thalamic (CBGT) circuit, sandwiched between striatal inputs and basal ganglia outputs. According to this model, one-way descending striatal signals in the indirect pathway amplify the suppression of downstream thalamic nuclei by inhibiting GPe activity. Here, we revisit this assumption, in light of new and emerging work on the cellular complexity, connectivity and functional role of the GPe in behaviour. We show how, according to this new circuit-level logic, the GPe is ideally positioned for relaying ascending and descending control signals within the basal ganglia. Focusing on the problem of inhibitory control, we illustrate how this bidirectional flow of information allows for the integration of reactive and proactive control mechanisms during action selection. Taken together, this new evidence points to the GPe as being a central hub in the CBGT circuit, participating in bidirectional information flow and linking multifaceted control signals to regulate behaviour.

A novel mechanism for ramping bursts based on slow negative feedback in model respiratory neurons.

Recordings from pre-Bötzinger complex neurons responsible for the inspiratory phase of the respiratory rhythm reveal a ramping burst pattern, starting around the time that the transition from expiration to inspiration begins, in which the spike rate gradually rises until a transition into a high-frequency burst occurs. The spike rate increase along the burst is accompanied by a gradual depolarization of the plateau potential that underlies the spikes. These effects may be functionally important for inducing the onset of inspiration and hence maintaining effective respiration; however, most mathematical models for inspiratory bursting do not capture this activity pattern. Here, we study how the modulation of spike height and afterhyperpolarization via the slow inactivation of an inward current can support various activity patterns including ramping bursts. We use dynamical systems methods designed for multiple timescale systems, such as bifurcation analysis based on timescale decomposition and averaging over fast oscillations, to generate an understanding of and predictions about the specific dynamic effects that lead to ramping bursts. We also analyze how transitions between ramping and other activity patterns may occur with parameter changes, which could be associated with experimental manipulations, environmental conditions, and/or development.

Slow negative feedback enhances robustness of square-wave bursting.

Square-wave bursting is an activity pattern common to a variety of neuronal and endocrine cell models that has been linked to central pattern generation for respiration and other physiological functions. Many of the reduced mathematical models that exhibit square-wave bursting yield transitions to an alternative pseudo-plateau bursting pattern with small parameter changes. This susceptibility to activity change could represent a problematic feature in settings where the release events triggered by spike production are necessary for function. In this work, we analyze how model bursting and other activity patterns vary with changes in a timescale associated with the conductance of a fast inward current. Specifically, using numerical simulations and dynamical systems methods, such as fast-slow decomposition and bifurcation and phase-plane analysis, we demonstrate and explain how the presence of a slow negative feedback associated with a gradual reduction of a fast inward current in these models helps to maintain the presence of spikes within the active phases of bursts. Therefore, although such a negative feedback is not necessary for burst production, we find that its presence generates a robustness that may be important for function.

Sensitivity to Control Signals in Triphasic Rhythmic Neural Systems: A Comparative Mechanistic Analysis via Infinitesimal Local Timing Response Curves.

Similar activity patterns may arise from model neural networks with distinct coupling properties and individual unit dynamics. These similar patterns may, however, respond differently to parameter variations and specifically to tuning of inputs that represent control signals. In this work, we analyze the responses resulting from modulation of a localized input in each of three classes of model neural networks that have been recognized in the literature for their capacity to produce robust three-phase rhythms: coupled fast-slow oscillators, near-heteroclinic oscillators, and threshold-linear networks. Triphasic rhythms, in which each phase consists of a prolonged activation of a corresponding subgroup of neurons followed by a fast transition to another phase, represent a fundamental activity pattern observed across a range of central pattern generators underlying behaviors critical to survival, including respiration, locomotion, and feeding. To perform our analysis, we extend the recently developed local timing response curve (lTRC), which allows us to characterize the timing effects due to perturbations, and we complement our lTRC approach with model-specific dynamical systems analysis. Interestingly, we observe disparate effects of similar perturbations across distinct model classes. Thus, this work provides an analytical framework for studying control of oscillations in nonlinear dynamical systems and may help guide model selection in future efforts to study systems exhibiting triphasic rhythmic activity.

Neural activity during a simple reaching task in macaques is counter to gating and rebound in basal ganglia-thalamic communication.

Task-related activity in the ventral thalamus, a major target of basal ganglia output, is often assumed to be permitted or triggered by changes in basal ganglia activity through gating- or rebound-like mechanisms. To test those hypotheses, we sampled single-unit activity from connected basal ganglia output and thalamic nuclei (globus pallidus-internus [GPi] and ventrolateral anterior nucleus [VLa]) in monkeys performing a reaching task. Rate increases were the most common peri-movement change in both nuclei. Moreover, peri-movement changes generally began earlier in VLa than in GPi. Simultaneously recorded GPi-VLa pairs rarely showed short-time-scale spike-to-spike correlations or slow across-trials covariations, and both were equally positive and negative. Finally, spontaneous GPi bursts and pauses were both followed by small, slow reductions in VLa rate. These results appear incompatible with standard gating and rebound models. Still, gating or rebound may be possible in other physiological situations: simulations show how GPi-VLa communication can scale with GPi synchrony and GPi-to-VLa convergence, illuminating how synchrony of basal ganglia output during motor learning or in pathological conditions may render this pathway effective. Thus, in the healthy state, basal ganglia-thalamic communication during learned movement is more subtle than expected, with changes in firing rates possibly being dominated by a common external source.

Identifying control ensembles for information processing within the cortico-basal ganglia-thalamic circuit.

In situations featuring uncertainty about action-reward contingencies, mammals can flexibly adopt strategies for decision-making that are tuned in response to environmental changes. Although the cortico-basal ganglia thalamic (CBGT) network has been identified as contributing to the decision-making process, it features a complex synaptic architecture, comprised of multiple feed-forward, reciprocal, and feedback pathways, that complicate efforts to elucidate the roles of specific CBGT populations in the process by which evidence is accumulated and influences behavior. In this paper we apply a strategic sampling approach, based on Latin hypercube sampling, to explore how variations in CBGT network properties, including subpopulation firing rates and synaptic weights, map to variability of parameters in a normative drift diffusion model (DDM), representing algorithmic aspects of information processing during decision-making. Through the application of canonical correlation analysis, we find that this relationship can be characterized in terms of three low-dimensional control ensembles within the CBGT network that impact specific qualities of the emergent decision policy: responsiveness (a measure of how quickly evidence evaluation gets underway, associated with overall activity in corticothalamic and direct pathways), pliancy (a measure of the standard of evidence needed to commit to a decision, associated largely with overall activity in components of the indirect pathway of the basal ganglia), and choice (a measure of commitment toward one available option, associated with differences in direct and indirect pathways across action channels). These analyses provide mechanistic predictions about the roles of specific CBGT network elements in tuning the way that information is accumulated and translated into decision-related behavior.

Rigorous Mapping of Data to Qualitative Properties of Parameter Values and Dynamics: A Case Study on a Two-Variable Lotka-Volterra System.

In this work, we describe mostly analytical work related to a novel approach to parameter identification for a two-variable Lotka-Volterra (LV) system. Specifically, this approach is qualitative, in that we aim not to determine precise values of model parameters but rather to establish relationships among these parameter values and properties of the trajectories that they generate, based on a small number of available data points. In this vein, we prove a variety of results about the existence, uniqueness, and signs of model parameters for which the trajectory of the system passes exactly through a set of three given data points, representing the smallest possible data set needed for identification of model parameter values. We find that in most situations such a data set determines these values uniquely; we also thoroughly investigate the alternative cases, which result in nonuniqueness or even nonexistence of model parameter values that fit the data. In addition to results about identifiability, our analysis provides information about the long-term dynamics of solutions of the LV system directly from the data without the necessity of estimating specific parameter values.

Transmission of delta band (0.5-4 Hz) oscillations from the globus pallidus to the substantia nigra pars reticulata in dopamine depletion.

Parkinson's disease (PD) and animal models of PD feature enhanced oscillations in several frequency bands in the basal ganglia (BG). Past research has emphasized the enhancement of 13-30 Hz beta oscillations. Recently, however, oscillations in the delta band (0.5-4 Hz) have been identified as a robust predictor of dopamine loss and motor dysfunction in several BG regions in mouse models of PD. In particular, delta oscillations in the substantia nigra pars reticulata (SNr) were shown to lead oscillations in motor cortex (M1) and persist under M1 lesion, but it is not clear where these oscillations are initially generated. In this paper, we use a computational model to study how delta oscillations may arise in the SNr due to projections from the globus pallidus externa (GPe). We propose a network architecture that incorporates inhibition in SNr from oscillating GPe neurons and other SNr neurons. In our simulations, this configuration yields firing patterns in model SNr neurons that match those measured in vivo. In particular, we see the spontaneous emergence of near-antiphase active-predicting and inactive-predicting neural populations in the SNr, which persist under the inclusion of STN inputs based on experimental recordings. These results demonstrate how delta oscillations can propagate through BG nuclei despite imperfect oscillatory synchrony in the source site, narrowing down potential targets for the source of delta oscillations in PD models and giving new insight into the dynamics of SNr oscillations.

Dynamics of ramping bursts in a respiratory neuron model.

Intensive computational and theoretical work has led to the development of multiple mathematical models for bursting in respiratory neurons in the pre-Bötzinger Complex (pre-BötC) of the mammalian brainstem. Nonetheless, these previous models have not captured the pre-inspiratory ramping aspects of these neurons' activity patterns, in which relatively slow tonic spiking gradually progresses to faster spiking and a full-blown burst, with a corresponding gradual development of an underlying plateau potential. In this work, we show that the incorporation of the dynamics of the extracellular potassium ion concentration into an existing model for pre-BötC neuron bursting, along with some parameter adjustments, suffices to induce this ramping behavior. Using fast-slow decomposition, we show that this activity can be considered as a form of parabolic bursting, but with burst termination at a homoclinic bifurcation rather than as a SNIC bifurcation. We also investigate the parameter-dependence of these solutions and show that the proposed model yields a greater dynamic range of burst frequencies, durations, and duty cycles than those produced by other models in the literature.

A data-driven model of the role of energy in sepsis.

Exposure to pathogens elicits a complex immune response involving multiple interdependent pathways. This response may mitigate detrimental effects and restore health but, if imbalanced, can lead to negative outcomes including sepsis. This complexity and need for balance pose a challenge for clinicians and have attracted attention from modelers seeking to apply computational tools to guide therapeutic approaches. In this work, we address a shortcoming of such past efforts by incorporating the dynamics of energy production and consumption into a computational model of the acute immune response. With this addition, we performed fits of model dynamics to data obtained from non-human primates exposed to Escherichia coli. Our analysis identifies parameters that may be crucial in determining survival outcomes and also highlights energy-related factors that modulate the immune response across baseline and altered glucose conditions.

The credit assignment problem in cortico-basal ganglia-thalamic networks: A review, a problem and a possible solution.

The question of how cortico-basal ganglia-thalamic (CBGT) pathways use dopaminergic feedback signals to modify future decisions has challenged computational neuroscientists for decades. Reviewing the literature on computational representations of dopaminergic corticostriatal plasticity, we show how the field is converging on a normative, synaptic-level learning algorithm that elegantly captures both neurophysiological properties of CBGT circuits and behavioral dynamics during reinforcement learning. Unfortunately, the computational studies that have led to this normative algorithmic model have all relied on simplified circuits that use abstracted action-selection rules. As a result, the application of this corticostriatal plasticity algorithm to a full model of the CBGT pathways immediately fails because the spatiotemporal distance between integration (corticostriatal circuits), action selection (thalamocortical loops) and learning (nigrostriatal circuits) means that the network does not know which synapses should be reinforced to favor previously rewarding actions. We show how observations from neurophysiology, in particular the sustained activation of selected action representations, can provide a simple means of resolving this credit assignment problem in models of CBGT learning. Using a biologically realistic spiking model of the full CBGT circuit, we demonstrate how this solution can allow a network to learn to select optimal targets and to relearn action-outcome contingencies when the environment changes. This simple illustration highlights how the normative framework for corticostriatal plasticity can be expanded to capture macroscopic network dynamics during learning and decision-making.

Delta oscillations are a robust biomarker of dopamine depletion severity and motor dysfunction in awake mice.

Delta oscillations (0.5-4 Hz) are a robust feature of basal ganglia pathophysiology in patients with Parkinson's disease (PD) in relationship to tremor, but their relationship to other parkinsonian symptoms has not been investigated. While delta oscillations have been observed in mouse models of PD, they have only been investigated in anesthetized animals, suggesting that the oscillations may be an anesthesia artifact and limiting the ability to relate them to motor symptoms. Here, we establish a novel approach to detect spike oscillations embedded in noise to provide the first study of delta oscillations in awake, dopamine-depleted mice. We find that approximately half of neurons in the substantia nigra pars reticulata (SNr) exhibit delta oscillations in dopamine depletion and that these oscillations are a strong indicator of dopamine loss and akinesia, outperforming measures such as changes in firing rate, irregularity, bursting, and synchrony. These oscillations are typically weakened, but not ablated, during movement. We further establish that these oscillations are caused by the loss of D2-receptor activation and do not originate from motor cortex, contrary to previous findings in anesthetized animals. Instead, SNr oscillations precede those in M1 at a 100- to 300-ms lag, and these neurons' relationship to M1 oscillations can be used as the basis for a novel classification of SNr into two subpopulations. These results give insight into how dopamine loss leads to motor dysfunction and suggest a reappraisal of delta oscillations as a marker of akinetic symptoms in PD.NEW & NOTEWORTHY This work introduces a novel method to detect spike oscillations amidst neural noise. Using this method, we demonstrate that delta oscillations in the basal ganglia are a defining feature of awake, dopamine-depleted mice and are strongly correlated with dopamine loss and parkinsonian motor symptoms. These oscillations arise from a loss of D2-receptor activation and do not require motor cortex. Similar oscillations in human patients may be an underappreciated marker and target for Parkinson's disease (PD) treatment.

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 (INaP), 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 INaP 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 INaP block, (2) show that the effects of INaP 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 INaP in the intact respiratory network, and (4) derive the prediction that effective block of INaP by low concentration tetrodotoxin will stop respiratory rhythm generation in the intact respiratory network. These simulations support a critical role for INaP in respiratory rhythmogenesis in vivo and illustrate the importance of considering mechanism when interpreting and simulating data relating to pharmacological blockade.

Robustness of respiratory rhythm generation across dynamic regimes.

A central issue in the study of the neural generation of respiratory rhythms is the role of the intrinsic pacemaking capabilities that some respiratory neurons exhibit. The debate on this issue has occurred in parallel to investigations of interactions among respiratory network neurons and how these contribute to respiratory behavior. In this computational study, we demonstrate how these two issues are inextricably linked. We use simulations and dynamical systems analysis to show that once a conditional respiratory pacemaker, which can be tuned across oscillatory and non-oscillatory dynamic regimes in isolation, is embedded into a respiratory network, its dynamics become masked: the network exhibits similar dynamical properties regardless of the conditional pacemaker node's tuning, and that node's outputs are dominated by network influences. Furthermore, the outputs of the respiratory central pattern generator as a whole are invariant to these changes of dynamical properties, which ensures flexible and robust performance over a wide dynamic range.

The roles of ascending sensory signals and top-down central control in the entrainment of a locomotor CPG.

Previous authors have proposed two basic hypotheses about the factors that form the basis of locomotor rhythms in walking insects: sensory feedback only or sensory feedback together with rhythmic activity of small neural circuits called central pattern generators (CPGs). Here we focus on the latter. Following this concept, to generate functional outputs, locomotor control must feature both rhythm generation by CPGs at the level of individual joints and coordination of their rhythmic activities, so that all muscles are activated in an appropriate pattern. This work provides an in-depth analysis of an aspect of this coordination process based on an existing network model of stick insect locomotion. Specifically, we consider how the control system for a single joint in the stick insect leg may produce rhythmic output when subjected to ascending sensory signals from other joints in the leg. In this work, the core rhythm generating CPG component of the joint under study is represented by a classical half-center oscillator constrained by a basic set of experimental observations. While the dynamical features of this CPG, including phase transitions by escape and release, are well understood, we provide novel insights about how these transition mechanisms yield entrainment to the incoming sensory signal, how entrainment can be lost under variation of signal strength and period or other perturbations, how entrainment can be restored by modulation of tonic top-down drive levels, and how these factors impact the duty cycle of the motor output.

Paying Our Dues: The Role of Professional Societies in the Evolution of Mathematical Biology Education.

Mathematical biology education provides key foundational underpinnings for the scholarly work of mathematical biology. Professional societies support such education efforts via funding, public speaking opportunities, Web presence, publishing, workshops, prizes, opportunities to discuss curriculum design, and support of mentorship and other means of sustained communication among communities of scholars. Such programs have been critical to the broad expansion of the range and visibility of research and educational activities in mathematical biology. We review these efforts, past and present, across multiple societies-the Society for Mathematical Biology (SMB), the Symposium on Biomathematics and Ecology Education and Research (BEER), the Mathematical Association of America (MAA), and the Society for Industrial and Applied Mathematics (SIAM). We then proceed to suggest ways that professional societies can serve as advocates and community builders for mathematical biologists at all levels, noting that education continues throughout a career and also emphasizing the value of educating new generations of students. Our suggestions include collecting and disseminating data related to biomath education; developing and maintaining mentoring systems and research communities; and providing incentives and visibility for educational efforts within mathematical biology.

Complex bursting dynamics in an embryonic respiratory neuron model.

Pre-Bötzinger complex (pre-BötC) network activity within the mammalian brainstem controls the inspiratory phase of the respiratory rhythm. While bursting in pre-BötC neurons during the postnatal period has been extensively studied, less is known regarding inspiratory pacemaker neuron behavior at embryonic stages. Recent data in mouse embryo brainstem slices have revealed the existence of a variety of bursting activity patterns depending on distinct combinations of burst-generating INaP and ICAN conductances. In this work, we consider a model of an isolated embryonic pre-BötC neuron featuring two distinct bursting mechanisms. We use methods of dynamical systems theory, such as phase plane analysis, fast-slow decomposition, and bifurcation analysis, to uncover mechanisms underlying several different types of intrinsic bursting dynamics observed experimentally including several forms of plateau bursts, bursts involving depolarization block, and various combinations of these patterns. Our analysis also yields predictions about how changes in the balance of the two bursting mechanisms contribute to alterations in an inspiratory pacemaker neuron activity during prenatal development.

Reduced computational modelling of Kölliker-Fuse contributions to breathing patterns in Rett syndrome.

KEY POINTS: Reduced computational models are used to test effects of loss of inhibition to the Kölliker-Fuse nucleus (KFn). Three reduced computational models that simulate eupnoeic and vagotomized respiratory rhythms are considered. All models exhibit the emergence of respiratory perturbations associated with Rett syndrome as inhibition to the KFn is diminished. Simulations suggest that application of 5-HT1A agonists can mitigate the respiratory pathology. The three models can be distinguished and tested based on their predictions about connections and dynamics within the respiratory circuit and about effects of perturbations on certain respiratory neuron populations. ABSTRACT: Rett syndrome (RTT) is a developmental disorder that can lead to respiratory disturbances featuring prolonged apnoeas of variable durations. Determining the mechanisms underlying these effects at the level of respiratory neural circuits would have significant implications for treatment efforts and would also enhance our understanding of respiratory rhythm generation and control. While experimental studies have suggested possible factors contributing to the respiratory patterns of RTT, we take a novel computational approach to the investigation of RTT, which allows for direct manipulation of selected system parameters and testing of specific hypotheses. Specifically, we present three reduced computational models, developed using an established framework, all of which successfully simulate respiratory outputs across eupnoeic and vagotomized conditions. All three models show that loss of inhibition to the Kölliker-Fuse nucleus reproduces the key respiratory alterations associated with RTT and, as suggested experimentally, that effects of 5-HT1A agonists on the respiratory neural circuit suffice to alleviate this respiratory pathology. Each of the models makes distinct predictions regarding the neuronal populations and interactions underlying these effects, suggesting natural directions for future experimental testing.

Mathematical modeling of energy consumption in the acute inflammatory response.

When a pathogen invades the body, an acute inflammatory response is activated to eliminate the intruder. In some patients, runaway activation of the immune system may lead to collateral tissue damage and, in the extreme, organ failure and death. Experimental studies have found an association between severe infections and depletion in levels of adenosine triphosphate (ATP), increase in nitric oxide production, and accumulation of lactate, suggesting that tissue energetics is compromised. In this work we present a differential equations model that incorporates the dynamics of ATP, nitric oxide, and lactate accompanying an acute inflammatory response and employ this model to explore their roles in shaping this response. The bifurcation diagram of the model system with respect to the pathogen growth rate reveals three equilibrium states characterizing the health, aseptic and septic conditions. We explore the domains of attraction of these states to inform the instantiation of heterogeneous virtual patient populations utilized in a survival analysis. We then apply the model to study alterations in the inflammatory response and survival outcomes in metabolically altered conditions such as hypoglycemia, hyperglycemia, and hypoxia.