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Spike time response curves (STRCs) are used to study the influence of synaptic stimuli on the firing times of a neuron oscillator without the assumption of weak coupling. They allow us to approximate the dynamics of synchronous state in networks of neurons through a discrete map. Linearization about the fixed point of the discrete map can then be used to predict the stability of patterns of synchrony in the network. General theory for taking into account the contribution from higher order STRC terms, in the approximation of the discrete map for coupled neuronal oscillators in synchrony is still lacking. Here we present a general framework to account for higher order STRC corrections in the approximation of discrete map to determine the domain of 1:1 phase locking state in the network of two interacting neurons. We begin by demonstrating that the effect of synaptic stimuli through a shunting synapse to a neuron firing in the gamma frequency band (20-80 Hz) last for three consecutive firing cycles. We then show that the discrete map derived by taking into account the higher order STRC contributions is successfully able predict the domain of synchronous 1:1 phase locked state in a network of two heterogeneous interneurons coupled through a shunting synapse.
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Axonal growth and guidance at the ventral floor plate is here followed $textit{in vivo}$ in real time at high resolution by light-sheet microscopy along several hundred micrometers of the zebrafish spinal cord. The recordings show the strikingly stereotyped spatio-temporal control that governs midline crossing. Commissural axons are observed crossing the ventral floor plate midline perpendicularly at about 20 microns/h, in a manner dependent on the Robo3 receptor and with a growth rate minimum around the midline, confirming previous observations. At guidance points, commissural axons are seen to decrease their growth rate and growth cones increase in size. Commissural filopodia appear to interact with the nascent neural network, and thereby trigger immediate plastic and reversible sinusoidal-shaped bending movements of neighboring commissural shafts. Ipsilateral axons extend concurrently, but straight and without bends, at three to six times higher growth rates than commissurals, indicating they project their path on a substrate-bound surface rather than relying on diffusible guidance cues. Growing axons appeared to be under stretch, an observation that is of relevance for tension-based models of cortical morphogenesis. The textit{in vivo} observations provide for a discussion of the current distinction between substrate-bound and diffusible guidance cues. The study applies the transparent zebrafish model that provides an experimental model system to explore further the cellular, molecular and physical mechanisms involved during axonal growth, guidance and midline crossing through a combination of $textit{in vitro}$ and $textit{in vivo}$ approaches.
Previous papers have developed a statistical mechanics of neocortical interactions (SMNI) fit to short-term memory and EEG data. Adaptive Simulated Annealing (ASA) has been developed to perform fits to such nonlinear stochastic systems. An N-dimensional path-integral algorithm for quantum systems, qPATHINT, has been developed from classical PATHINT. Both fold short-time propagators (distributions or wave functions) over long times. Previous papers applied qPATHINT to two systems, in neocortical interactions and financial options. textbf{Objective}: In this paper the quantum path-integral for Calcium ions is used to derive a closed-form analytic solution at arbitrary time that is used to calculate interactions with classical-physics SMNI interactions among scales. Using fits of this SMNI model to EEG data, including these effects, will help determine if this is a reasonable approach. textbf{Method}: Methods of mathematical-physics for optimization and for path integrals in classical and quantum spaces are used for this project. Studies using supercomputer resources tested various dimensions for their scaling limits. In this paper the quantum path-integral is used to derive a closed-form analytic solution at arbitrary time that is used to calculate interactions with classical-physics SMNI interactions among scales. textbf{Results}: The mathematical-physics and computer parts of the study are successful, in that there is modest improvement of cost/objective functions used to fit EEG data using these models. textbf{Conclusion}: This project points to directions for more detailed calculations using more EEG data and qPATHINT at each time slice to propagate quantum calcium waves, synchronized with PATHINT propagation of classical SMNI.
Seizure activity is a ubiquitous and pernicious pathophysiology that, in principle, should yield to mathematical treatments of (neuronal) ensemble dynamics - and therefore interventions on stochastic chaos. A seizure can be characterised as a deviation of neural activity from a stable dynamical regime, i.e. one in which signals fluctuate only within a limited range. In silico treatments of neural activity are an important tool for understanding how the brain can achieve stability, as well as how pathology can lead to seizures and potential strategies for mitigating instabilities, e.g. via external stimulation. Here, we demonstrate that the (neuronal) state equation used in Dynamic Causal Modelling generalises to a Fokker-Planck formalism when propagation of neuronal activity along structural connections is considered. Using the Jacobian of this generalised state equation, we show that an initially unstable system can be rendered stable via a reduction in diffusivity (i.e., connectivity that disperses neuronal fluctuations). We show, for neural systems prone to epileptic seizures, that such a reduction can be achieved via external stimulation. Specifically, we show that this stimulation should be applied in such a way as to temporarily mirror epileptic activity in the areas adjoining an affected brain region - thus fighting seizures with seizures. We offer proof of principle using simulations based on functional neuroimaging data collected from patients with idiopathic generalised epilepsy, in which we successfully suppress pathological activity in a distinct sub-network. Our hope is that this technique can form the basis for real-time monitoring and intervention devices that are capable of suppressing or even preventing seizures in a non-invasive manner.
We consider a pair of stochastic integrate and fire neurons receiving correlated stochastic inputs. The evolution of this system can be described by the corresponding Fokker-Planck equation with non-trivial boundary conditions resulting from the refractory period and firing threshold. We propose a finite volume method that is orders of magnitude faster than the Monte Carlo methods traditionally used to model such systems. The resulting numerical approximations are proved to be accurate, nonnegative and integrate to 1. We also approximate the transient evolution of the system using an Ornstein--Uhlenbeck process, and use the result to examine the properties of the joint output of cell pairs. The results suggests that the joint output of a cell pair is most sensitive to changes in input variance, and less sensitive to changes in input mean and correlation.
One of the main causes of long-term prosthetic abandonment is the lack of ownership over the prosthesis, caused mainly by the absence of sensory information regarding the lost limb. One strategy to overcome this problem is to provide alternative feedback mechanisms to convey information respective to the absent limb. To address this issue, we developed a Biofeedback system for the rehabilitation of transfemoral amputees, controlled via electromyographic activity from the leg muscles, that can provide real-time visual and/or vibratory feedback for the user. In this study, we tested this device with able-bodied individuals performing an adapted version of the clinical protocol. Our idea was to test the effectiveness of combining vibratory and visual feedbacks and how task difficulty affects overall performance. Our results show no negative interference combining both feedback modalities, and that performance peaked at the intermediate difficulty. These results provide powerful insights of what can be expected with the population of amputee people and will help in the final steps of protocol development. Our goal is to use this biofeedback system to engage another sensory modality in the process of spatial representation of a virtual leg, bypassing the lack of information associated with the disruption of afferent pathways following amputation.
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