Biological Cybernetics

A novel mathematical framework to define the threshold of action potentials in excitable cells is presented. Unlike previously applied methods that rely on approximations or bifurcations, the approach focuses on the geometry of membrane potential trajectories. The changes in concavity during the upstroke of an action potential can be directly obtained from a time series of voltages. The concavity criterion is then extended to models based on autonomous dynamical systems where the changes in concavity can be obtained analytically from a curve of inflection points in phase space. The inflection point manifold defines a region required for excitability: all the orbits that cross it contain action potentials, and all the trajectories that contain action potentials are in it. This analytical principle can then be used to define excitability in a dynamical system, and also a measure of excitability that enables quantification and comparisons of excitability across dynamical system. The measure provides a way to compare the excitabilities of systems that model neurons with different electrophysiological phenotypes and consider different stimulus conditions. The traditionally vague physiological concept of electrical excitability is transformed into a rigorous analytical description by considering the time-dependent curvature of the membrane potential. The criterion is robust across smooth, single compartment models of electrical excitability and can be can be extended to single compartment models in higher dimensions, and multicompartment models as well.

Animals use long-range signals, such as hormones and neural signals, to coordinate the actions of distant organs. There is no precise, quantitative framework that explains the problems these control systems must solve and thus predicts their behavior under varied conditions. We consider this problem in the context of blood glucose regulation by the hormone insulin, the failure of which produces diabetes. We show that existing mathematical models of glucose regulation admit equivalent control strategies with no hormones at all, and thus cannot explain the need for hormonal regulation. We therefore introduce a minimal model of inter-organ variations in local glucose, and show that control strategies based on local glucose measurements face severe trade-offs between different control objectives. In contrast, we show that hormonal control signals from the pancreas can overcome these limitations. By exposing the benefits of hormonal control, our work paves the way to a detailed understanding of physiological design principles, with possible implications for the engineering of an artificial pancreas.

Oscillations and oscillatory synchronization are pervasive in motor circuits, where their role in rhythm generation and entrainment is well established but their role in feedback control of movement remains unclear. Here I show analytically that two oscillators of any type, coupled through a delayed interaction that is an odd function of their phase difference, necessarily implement a proportional-derivative (PD) control law in the near-synchrony limit. The proportional gain follows from the slope of the coupling function and the derivative gain is set by the coupling delay, so that PD control emerges with no additional machinery. Simulations confirm that such oscillators reproduce ideal PD step responses near synchrony and that control quality degrades systematically away from it. This establishes a direct, model-independent bridge between oscillatory synchronization and feedback control, and suggests concrete experimental signatures for candidate systems.

Length and time constants are foundational to the study of conduction in neurons and other biological cables but are exactly defined only for passive membranes. Here we define and derive exact length and time constants for propagating action potentials in unmyelinated axons. This derivation exploits specific instants during action potential conduction when the net transmembrane ionic current is zero, but axial current remains non-zero. At these instants, we define a curvature parameter,{kappa} , explore its determinants using computer modelling, demonstrate that it is the local real Laplace exponent of the action potential upstroke, and suggest practical approaches for its experimental measurement. From{kappa} , we define action potential length and time constants, {lambda}AP = 1/{surd}({kappa}racm) and {tau}AP = 1/{kappa}, and show that action potential propagation velocity is exactly {lambda}AP/{tau}AP.

Pacemaking neurons, often found in mammalian nervous systems, integrate their inputs differently than quiescent neurons. Rhythmic single-spike pacemaking that is robust to noise can be achieved with a slow process that enforces a "resting potential" at each point along a ramp-like interspike interval (ISI) coupled with a fast restorative component. To demonstrate this phenomenon, we modeled previously identified distinct subpopulations of midbrain dopamine neurons that differed in projection target and in the regularity of their pacemaking. In the model of the more regularly-firing subpopulation projecting to the dorsomedial striatum, KV4 current was recruited by a deep after-hyperpolarizing potential (AHP) mediated by the SK channel. In the model of the less regularly-firing subpopulation projecting to the medial shell of the nucleus accumbens, the AHP was too shallow to recruit the KV4 current. In the more regularly firing population, the trajectory in the phase space of membrane potential and slow inactivation of KV4 was confined to move slowly through a narrow channel during the ramp-like portion of the ISI. Noisy perturbations from this channel were quickly damped by fast activation of KV4. In contrast, the smaller AHP in the model of the subpopulation projecting to the medial shell of the nucleus accumbens failed to recruit Kv4-mediated current, therefore the narrow channel was never entered, greatly decreasing the regularity in the presence of noise. This mechanism may be broadly applicable to single-spike pacemakers and explains how slow pacemaking with small net currents can be robust to fluctuations in single channel openings. Author SummaryPacemaking cells spike at regular intervals without the need for external input. There are numerous examples of pacemaking cells in the nervous system. We show that a process with slow dynamics relative to the individual spikes can make regular pacemaking robust to the noise that is always present in biological systems.

This study presents improvements to the Hodgkin-Huxley (HH) models of ionic conductance and action potential generation. Sodium and potassium conductances are expressed by a single analytical formula describing the impulse response of a convolution of exponential distributions within a short-memory integration space. Treating transmembrane ion transit duration as a random variable, conductance profiles are interpreted as realizations of the probability density functions governing ionic movements. Applying the central limit theorem, the lognormal distribution emerges as the asymptotic profile of ionic conductances, constituting a fundamental primitive for such biosignals. A temporal state-transition paradigm describes the action potential waveform through four successive membrane potential transitions. Applied to electrophysiological recordings from lamprey reticulospinal neurons, this framework enables indirect estimation of key physiological quantities, including depolarization threshold, Nernst potentials, and net ion fluxes across the membrane. These advances open new perspectives for parameter estimation from experimental data and neuronal network simulation.

Stabilizing synaptic plasticity together with the neurons activity has remained a central challenge in theoretical neuroscience since the introduction of Hebbian learning principles. Classical Hebbian learning rules typically lead to unbounded synaptic growth, motivating the development of stabilization mechanisms such as normalization methods, BCM-type learning, synaptic scaling and others. While these approaches can prevent divergence, they can also exhibit different limitations e.g. resulting in too-sparse synaptic configurations or leading to poor scalability with increasing network size. A recently introduced meta-plasticity mechanism, termed annealed linear learning (ALL), dynamically reduces the learning rate as neuronal output increases, thereby preserving stable and interpretable fixed-point behavior of the output. However, the original formulation leads to an irreversible decay of the learning rate, preventing adaptation to changing environmental conditions. To address this, in the present study, we balance learning rate reduction at large outputs with recovery at small outputs and in addition introduce forgetting that gradually reduces synaptic weights. These extensions allow the system to discard outdated representations and adapt to novel input conditions. Analytical investigations demonstrate that the favorable output fixed-point properties of the original ALL framework are preserved under the extended rule. Furthermore, simulations with an artificial agent show that the proposed mechanism enables robust and fast re-learning and adaptation in changing environments.

With 302 neurons and a rigorously characterized connectome, the nematode Caenorhabditis elegans represents a powerful model organism to study the fundamental roles of neuronal circuits in behavior. However, despite the breadth of research, many questions remain unanswered regarding how these organisms are able to successfully navigate their environment. Here, we present a biologically grounded dynamical circuit model for the investigation of sensory-guided behavior during C. elegans chemotaxis. Our mathematical model consists of the chemosensory neuron AWA, interneurons RIM and RIA, motor neurons, including SMDs and RMDs, and body wall muscles that provide proprioceptive feedback through stretch receptors. After optimization with an evolutionary algorithm, the model locomotes effectively toward a chemical attractant, realistically capturing nematode chemotactic behavior. Chemotaxis is ensured by sharp turns, which resemble the omega turns of living nematodes, as a key emergent property of the model. The sharp turning behavior is triggered by decreases in the concentration of the attractant. These result in reduced AWA activity, which in turn triggers disinhibition of RIM and subsequent changes in RIA oscillations. The ensuing coordinated changes in downstream motor neurons activity patterns produce sharp turns, which correct the nematodes path, so that the model worm heads toward the attractant, and remains at its proximity, after it reaches the gradient peak. The proposed framework, along with its emergent dynamics, provides new insights into the minimum requirements for C. elegans circuitry to display major features of its chemotactic behavior, including omega turns. In parallel, it generates experimentally testable hypotheses with respect to the participating neuronal elements.

Human quiet standing depends on the context-dependent reweighting of vestibular, proprioceptive, and visual information. Posturography has empirically characterized this phenomenon, but a minimal generative-control account of how changes in sensory reliability propagate from state estimation to postural action remains incomplete. Here, we test a minimal continuous-time active inference model of quiet standing. In this model, sensory reweighting is implemented as channel-specific precision control over prediction errors. A one-link inverted pendulum receives vestibular, proprioceptive, and visual observations, estimates posture using a generalized-coordinate variational free-energy objective, and selects ankle torque by minimizing the same objective under an upright generalized sensory goal. Context changes alter the relative precision of sensory channels, without changing the plant, action optimizer, or goal dynamics. Across controlled sensory perturbations, reducing the precision of an unreliable visual or proprioceptive channel reduced perturbation-driven postural shifts by approximately 82%. An automatic-differentiation-based state-update gradient contribution closely matched the reduction, identifying the mechanistic locus of reweighting in the perceptual update. A linear reliability-to-precision law, lambda(c) = 1 + 7c, monotonically controlled sensory contribution, and a fixed-precision ablation showed that global precision reduction did not produce reweighting: the behavioral bias remained comparable to the equal-precision condition unless precision was changed selectively across channels. These results support the claim that postural sensory reweighting can be understood as relative, context-selective precision control in a continuous active inference loop.

Behaviors and thoughts are driven by a multitude of nested neuronal circuitry loops. They cause complex brain activity dynamics that remain poorly understood. We show that closed-loop neuronal network operation results in an activity state space that can be best understood as a vector field with an attractor point, which controls the activity dynamics across the neuronal population. We show that brain activity in vivo, however, indicates the attractor point is continually moving along a trajectory, which requires the presence of dynamic sensory input or independent activity generation within neurons. Using a spinal network model receiving sensory feedback from a dynamical biomechanical system, we show how these two independent dynamical systems mutually drive each others activity trajectories to generate behavior. Similarly, independent self-generated activity within each thalamic neuron, in closed loop with cortical subpopulations, results in a multitude of dynamical subnetworks that shape each others activity trajectories to control cortical populations. Although the attractor trajectories reflect emergent stability, we show them to be susceptible to criticality effects where minor changes in synaptic inputs can cause the attractor trajectory to switch to cause alternative behaviors. This renders the mutual perturbations between neural and biomechanical dynamics, and between subnetworks within the CNS, an effective operational mode to achieve behavioral flexibility and to simplify learning of apparently complex behaviors. We illustrate how this mode of operation necessitates anticipatory control, thoughts, by the cortex and discuss how it can encompass also the other CNS structures involved in somatic sensorimotor control.

Neuromechanical simulations provide a powerful framework for investigating how neural control architectures generate and regulate human locomotion. Numerous biologically inspired locomotion controllers have been proposed, including reflex-based, central pattern generator (CPG)-based, and muscle synergy-based models. However, direct comparison across studies remains difficult because of differences in musculoskeletal models, optimization methods, and evaluation protocols. Here, we implemented four representative locomotion control architectures, reflex-based, CPG-reflex-based, muscle synergy-based, and CPG-reflex-synergy-based controllers, within a unified neuromechanical simulation framework to enable controlled comparisons under shared biomechanical and computational conditions. Performance was assessed in terms of (1) agreement with experimentally observed gait characteristics, including kinematics, kinetics, muscle activations, and biomechanical trends across speeds and slopes, and (2) locomotor versatility across speed-slope conditions. The reflex-based and CPG-reflex-synergy-based controllers most closely reproduced experimentally observed gait characteristics, while the CPG-reflex-synergy controller achieved the broadest range of stable walking behaviors across speeds and slopes, followed closely by the reflex-based controller. These findings should be interpreted as comparisons of specific model implementations rather than definitive evaluations of the underlying biological hypotheses. Moreover, because the investigated controllers primarily focused on spinal-level mechanisms for nominal steady-state locomotion, the limited versatility observed in some of the models across broader speed and slope conditions suggests the importance of integrating spinal locomotor mechanisms with supraspinal modulation when modeling locomotion beyond nominal steady gait. To facilitate further investigation, we publicly share the simulation framework and controller implementations. Key pointsO_LIExisting neuromechanical locomotion controllers have been difficult to compare directly because of differences in simulation frameworks. C_LIO_LIWe implemented four representative spinal locomotion control models (reflex-based, central pattern generator (CPG)-reflex-based, muscle synergy-based, and CPG-reflex-synergy-based) within a unified simulation framework and compared their human-likeness and versatility. C_LIO_LIThe reflex-based and CPG-reflex-synergy-based controllers best reproduced human-like gait characteristics, while the CPG-reflex-synergy-based controller demonstrated the greatest locomotor versatility across speed-slope conditions, followed closely by the reflex-based controller. C_LIO_LIBecause the investigated controllers primarily modeled spinal-level mechanisms associated with steady-state locomotion, their reduced adaptability across broader speed and slope conditions highlights the importance of incorporating supraspinal modulation when modeling locomotion beyond nominal gait. C_LIO_LIWe publicly share the simulation framework and controller implementations to support further investigation of human locomotion control. C_LI

Natural physical environments constantly fluctuate across multiple timescales, often following a scale-free (1/f ) pattern where = 0.5 governs the fractional adaptation dynamics (Drew and Abbott 2006, Lundstrom et al. 2008). Here, we demonstrate how a multi-timescale sensory model successfully tracks these long-term trends to maintain stable encoding. Using an event-based Generalized Leaky Integrate-and-Fire (GLIF) paradigm, we found that a fast-adapting, single-exponential model with a short time constant{tau} [≤] 31.6 ms quickly crashes into complete refractory saturation when faced with large, low-frequency environmental shifts. In contrast, introducing a deep fractional memory tail of 1000.0 ms acts as an automated, high-pass balancing mechanism that continuously tracks and subtracts slow environmental variance. This predictive balancing prevents sensory collapse, anchors the mean firing rate to a steady homeostatic baseline, and maximizes coding efficiency for rapid, localized signals. Our results show that while a simple single-pole exponential model fails to retain history, a parallel bank of physiological relaxation processes converging on a target fractional profile t-0.5 provides the necessary historical memory to safely navigate natural stimulus fluctuations. Comfortingly, even a simplified three-pole approximation captures the bulk of this homeostatic benefit, making efficient fractional adaptation biologically viable at the sensory periphery without requiring infinite historical storage.

Predictions of skeletal muscle energy consumption under a diverse range of muscle contractile conditions are critical for improving our understanding of locomotion. Existing mathematical models, while capturing the mechanical dependence of energy consuming processes, neglect the time-dependent behaviour and recovery costs associated with regenerating ATP. This time-dependence is important for predicting the energetic response of muscles during repetitive or cyclical tasks like locomotion, where muscle undergoes many contraction cycles. This study presents a novel model to predict energetic rates based on physiological processes: Ca2+ transport costs, cross-bridge cycling costs, and ATP regeneration. Previous mathematical models include the dependence on Ca2+ transport and cross-bridge cycling, but neglect the time-dependent response and the subsequent recovery of ATP following the contraction. Model parameters were obtained from existing data on isolated muscle preparations, and predicted energetic rates were validated on separate datasets across a range of contractile conditions including dynamic, sub-maximal, and twitch contractions. The time-dependent model was able to capture the influence of contraction frequency on peak energetic rates and the time-course of energetic recovery observed experimentally. The model captures key physiological processes while maintaining a minimal number of free parameters and low computational cost. This enables generalisability across muscles and species, and implementation into larger scale musculoskeletal models.

Neural temporal point processes (NTPPs) are powerful tools for modeling sequences of timestamped events with statistical temporal structure. Density-based NTPPs, in particular, are an interesting opportunity to merge the universal function approximation capability of neural networks with a defined statistical model in a way that has many potential applications. We demonstrate one such application to heartbeat dynamics, a physiologic point process. We specifically apply a lognormal mixture NTPP to compute instantaneous estimates of the mean and standard deviation of beat-to-beat intervals. We compare our results to the state of art (Barbieri et al.) point process model for heartbeat dynamics, which uses a more physiologically rigorous inverse Gaussian model. We find that the NTPP model maintains reasonable accuracy while improving upon robustness to noise.

The activity of neural populations typically encodes more information about sensory or motor variables than can be captured by point estimates of the variables. We present and compare two approaches to quantifying this additional or ancillary information and its relationship to uncertainty: the mutual information between activity and estimation error, and the Fisher information loss, which can be interpreted in terms of curvature in information geometry. We show that deviations from Gaussianity of estimation errors, including the long tails frequently observed in human behavioural tasks, are an expected corollary of the presence of ancillary information. However, populations with similar distributions of estimation error can differ substantially in their ancillary information content depending on the noise characteristics. For a given population tuning and noise model, our results quantify an upper bound on the information about uncertainty that can be obtained from population activity alone: behaviour demonstrating knowledge in excess of this bound would indicate access to a separate source of information about uncertainty. Finally, we contrast the effects of external noise and decreasing internal signal strength on ancillary information and the Gaussianity of errors. Our work directly relates knowledge about uncertainty to non-Gaussianity in sensory estimates, and establishes a coherent theoretical foundation for investigating the basis of metacognition in neural population activity. Author summaryThe brain processes sensory evidence about the external world via inherently noisy neural activity. As a result, behavioural judgments - such as estimating the direction of a moving object - are fundamentally uncertain. While animals, including humans, routinely use uncertainty to guide decisions under risk, how neural populations represent this uncertainty remains unclear. In this work, we show how the same neural activity used to decode a sensory variable can also provide information about the estimates reliability. We introduce a mathematical framework to quantify this "ancillary information" directly from a neural populations encoding model. We demonstrate that ancillary information predicts non-Gaussianity in estimation errors and sets an upper bound on metacognitive sensitivity (how accurately subjective confidence tracks performance). Crucially, we show that neural populations with distinct noise characteristics can yield near-identical estimation errors while providing very different degrees of uncertainty information. This highlights the importance of evaluating ancillary information, not just error patterns, when comparing competing models of sensory coding.

Functional Electrical Stimulation (FES) therapy is a widely used neurorehabilitation technique that restores motor function by delivering electrical stimulation to target muscles during voluntary contraction. Despite its clinical effectiveness, the mechanisms by which FES therapy induces neuroplasticity remain poorly understood. Previous work has proposed that positive plasticity arises from Hebbian interactions at corticospinal-motoneuronal synapses when voluntary descending motor commands coincide with antidromic firing elicited by FES therapy. However, if spike-timing-dependent plasticity (STDP) is assumed to underlie this Hebbian mechanism, an unresolved question remains: why does FES therapy produce long-term potentiation (LTP) reliably, rather than the mixture of LTP and LTD predicted from classical STDP rules? Here, we test the hypothesis that interactions between voluntary descending spikes and stimulation-evoked antidromic spikes generate multi-spike patterns that bias plasticity toward potentiation. To investigate this mechanism, we developed a computational framework implementing a voltage-dependent plasticity rule that incorporates postsynaptic membrane dynamics and higher-order spike interactions. This framework enables simulation of synaptic plasticity during FES therapy while systematically varying stimulation frequency, input heterogeneity, and spike timing structure. Our simulations show that voltage-dependent dynamics strongly bias synaptic changes toward LTP during FES therapy-like conditions. In particular, physiological interspike interval variability promotes potentiation, whereas highly regular inputs bias synapses toward depression. These results indicate that postsynaptic voltage dynamics and spike-interaction structure, rather than pairwise spike timing alone, govern plasticity outcomes during FES therapy. Our findings provide a mechanistic explanation for why FES therapy reliably induces LTP-dominant plasticity and offer a computational framework for optimizing neuromodulation therapies.

Cortical populations exhibit a wide range of tuning properties, raising the question of whether such variability is a feature or a bug of cortical function. Prior work has shown that tuning diversity can improve population codes by mitigating the effects of correlated noise and increasing the discrimination and identification capacity of geometric representations. Motivated by these findings, we study a model in which a heterogeneous family of tuning curves, coding for a circular variable, is replicated at equally spaced preferred angles. We show that this heterogeneous population achieves better discrimination and detection than an equally sized homogeneous population constructed from shifted copies of the family's mean tuning curve, while using the same spike budget. Thus, homogeneous tuning is unstable under perturbations that preserve the mean tuning curve, because such perturbations leave metabolic cost unchanged while improving coding performance. We propose that such instability creates evolutionary pressure toward heterogeneity of tuning, making its prevalence a consequence of a process that optimizes coding performance under metabolic constraints.

Songbirds, in particular zebra finches (Taeniopygia guttata), provide a powerful model for investigating the neural mechanisms of learned vocal behavior. Researchers typically rely on the acoustic structure of birdsong to quantify vocal behavior. As a more direct measure of motor control, we present VIBE: Vocal acoustic Inversion to Biomechanical Estimates, an open-source pipeline that recovers the biomechanical control parameters of song production directly from the acoustic waveform. Biomechanical models of the songbird syrinx describe vocal production with two continuously varying parameters: and {beta}, representing subsyringeal air sac pressure and syringeal muscle tension, respectively. Recovering these parameters from song acoustics provides a motor-based coordinate system against which neural activity or other dependent variables can be directly compared. Because and {beta} are the coupled control parameters of a nonlinear oscillator, their joint recovery is non-trivial. VIBE addresses this through iterative optimization of the governing normal-form equations. We validate VIBE against recorded air sac pressure across 44 songs from twelve birds, showing that the recovered corresponds to empirically measured air sac pressure. Pairing VIBE with Neuropixels recordings from RA in five birds, we find that RA activity is well predicted by the recovered parameters, and that and {beta} add predictive power beyond the acoustic features of song. By recovering biomechanical control parameters from the acoustic signal, VIBE makes the biomechanical coordinate system of song production accessible to the broader songbird research community. New & NoteworthyVIBE provides a novel, fully automated pipeline to recover the biomechanical control parameters of the avian vocal organ, and {beta}, as continuously varying quantities from the raw acoustic waveform, making the full biomechanical model of song production accessible at the scale of modern datasets.

This paper introduces a sensorimotor learning framework for a corticocerebellar network, grounded in the perspective of population dynamics. Using an optimal control theory approach, the cerebellum model enhances preparatory activity through premotor input, allowing the motor cortex to reach the desired initial conditions for movement more efficiently. Unlike traditional motor learning approaches that focus on acquiring new skills, this paradigm emphasizes automatization of already executable behaviors through repetition driven by intrinsic motivation. The proposed model is evaluated using a center-out reaching task, demonstrating that the role of the cerebellum is to shorten the preparatory period required for the successful execution of the movement. These findings suggest that corticocerebellar interactions play a crucial role in optimizing motor preparation, offering insight into the neural mechanisms underlying movement efficiency.

Classical neurogeometric models describe the primary visual cortex as a fibered structure in which retinal position and local orientation are coupled through the geometry of the roto-translation group. We extend this approach to the visuomotor cortex by modeling it as an assemblage of visual and motor cortical geometries. The model combines orientation-selective representations, analogous to those of the primary visual cortex, with movement-direction-selective representations, analogous to those of the primary motor cortex, in order to describe the mixed visual and motor selectivity observed in the visuomotor cortex. We introduce a coupled visuomotor structure in which visual orientation and motor direction coexist over a common spatial plane and interact through a relative-orientation constraint. Neural responses are modeled by orientation- and direction-dependent profile functions, and preference maps are obtained from vectorized population responses. Numerical simulations generate visual, motor, and mixed visuomotor response maps. A competition rule between visual and motor responses produces incidence ratios close to experimental observations in macaque visuomotor cortex. This framework provides a first neurogeometric approximation of visuomotor functional architecture and a mathematical setting for studying visually guided action.