1,640 research outputs found

    Macroscopic equations governing noisy spiking neuronal populations

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    At functional scales, cortical behavior results from the complex interplay of a large number of excitable cells operating in noisy environments. Such systems resist to mathematical analysis, and computational neurosciences have largely relied on heuristic partial (and partially justified) macroscopic models, which successfully reproduced a number of relevant phenomena. The relationship between these macroscopic models and the spiking noisy dynamics of the underlying cells has since then been a great endeavor. Based on recent mean-field reductions for such spiking neurons, we present here {a principled reduction of large biologically plausible neuronal networks to firing-rate models, providing a rigorous} relationship between the macroscopic activity of populations of spiking neurons and popular macroscopic models, under a few assumptions (mainly linearity of the synapses). {The reduced model we derive consists of simple, low-dimensional ordinary differential equations with} parameters and {nonlinearities derived from} the underlying properties of the cells, and in particular the noise level. {These simple reduced models are shown to reproduce accurately the dynamics of large networks in numerical simulations}. Appropriate parameters and functions are made available {online} for different models of neurons: McKean, Fitzhugh-Nagumo and Hodgkin-Huxley models

    Multistable jittering in oscillators with pulsatile delayed feedback

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    Oscillatory systems with time-delayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in the last years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. In the bifurcation point numerous regimes with non-equal interspike intervals emerge simultaneously. We show that this bifurcation is triggered by the steepness of the oscillator's phase resetting curve and that the number of the emerging, so-called "jittering" regimes grows exponentially with the delay value. Although this appears as highly degenerate from a dynamical systems viewpoint, the "multi-jitter" bifurcation occurs robustly in a large class of systems. We observe it not only in a paradigmatic phase-reduced model, but also in a simulated Hodgkin-Huxley neuron model and in an experiment with an electronic circuit

    Bistable dynamics underlying excitability of ion homeostasis in neuron models

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    When neurons fire action potentials, dissipation of free energy is usually not directly considered, because the change in free energy is often negligible compared to the immense reservoir stored in neural transmembrane ion gradients and the long-term energy requirements are met through chemical energy, i.e., metabolism. However, these gradients can temporarily nearly vanish in neurological diseases, such as migraine and stroke, and in traumatic brain injury from concussions to severe injuries. We study biophysical neuron models based on the Hodgkin-Huxley (HH) formalism extended to include time-dependent ion concentrations inside and outside the cell and metabolic energy-driven pumps. We reveal the basic mechanism of a state of free energy-starvation (FES) with bifurcation analyses showing that ion dynamics is for a large range of pump rates bistable without contact to an ion bath. This is interpreted as a threshold reduction of a new fundamental mechanism of 'ionic excitability' that causes a long-lasting but transient FES as observed in pathological states. We can in particular conclude that a coupling of extracellular ion concentrations to a large glial-vascular bath can take a role as an inhibitory mechanism crucial in ion homeostasis, while the Na+^+/K+^+ pumps alone are insufficient to recover from FES. Our results provide the missing link between the HH formalism and activator-inhibitor models that have been successfully used for modeling migraine phenotypes, and therefore will allow us to validate the hypothesis that migraine symptoms are explained by disturbed function in ion channel subunits, Na+^+/K+^+ pumps, and other proteins that regulate ion homeostasis.Comment: 14 pages, 8 figures, 4 table
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