Optimal Inputs for Phase Models of Spiking Neurons
01 Oct 2006-Journal of Computational and Nonlinear Dynamics (American Society of Mechanical Engineers)-Vol. 1, Iss: 4, pp 358-367
TL;DR: It is shown that, for a given reduced neuron model and target spike time, there is a unique current that minimizes a squareintegral measure of its amplitude, which reflects the role of intrinsic neural dynamics in determining the time course of synaptic inputs to which a neuron is optimally tuned to respond.
Abstract: Variational methods are used to determine the optimal currents that elicit spikes in various phase reductions of neural oscillator models. We show that, for a given reduced neuron model and target spike time, there is a unique current that minimizes a squareintegral measure of its amplitude. For intrinsically oscillatory models, we further demonstrate that the form and scaling of this current is determined by the model’s phase response curve. These results reflect the role of intrinsic neural dynamics in determining the time course of synaptic inputs to which a neuron is optimally tuned to respond, and are illustrated using phase reductions of neural models valid near typical bifurcations to periodic firing, as well as the Hodgkin-Huxley equations. DOI: 10.1115/1.2338654 Phase-reduced models of neurons have traditionally been used to investigate either the patterns of synchrony that result from the type and architecture of coupling 1–8 or the response of large groups of oscillators to external stimuli 9–11. In all of these cases, the inputs to the model cells were fixed by definition of the model at the outset and the dynamics of phase models of networks or populations were analyzed in detail. The present paper takes a complementary, control-theoretic approach that is related to probabilistic “spike-triggered” methods 12: we fix at the outset a feature of the dynamical trajectories of interest—spiking at a precise time t1—and study the neural inputs that lead to this outcome. By computing the optimal such input, according to a measure of the input strength required to elicit the spike, we identify the signal to which the neuron is optimally “tuned” to respond. We view the present work as part of the first attempts 13,14 to understand the dynamical response of neurons using control theory, and, as we expect that insights from this general perspective will be combined with the “forward” dynamics results that Phil Holmes and many others have derived to ultimately enhance our understanding of neural processing, we hope that it will serve as a fitting tribute to his work.
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TL;DR: In this paper, the authors describe the rules of the ring, the ring population, and the need to get off the ring in order to measure the movement of a cyclic clock.
Abstract: 1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects
3,424 citations
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TL;DR: In this paper, the authors derive necessary and sufficient controllability conditions and an analytical optimal control law for an ensemble of finite-dimensional time-varying linear systems with the same control signal.
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TL;DR: It is shown that the type of PRC in cortical pyramidal neurons can be switched by cholinergic neuromodulation from type II (biphasic) to type I (monophasic) in a recent experimental study.
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98 citations
Cites methods from "Optimal Inputs for Phase Models of ..."
...In the case of a Hopf-Andronov bifurcation, the PRC is biphasic, or type II, with a phase delay in response to perturbations early in the period and a phase advance in response to perturbations late in the period (Hansel and Mato 1995; Ermentrout 1996; Moehlis et al. 2006)....
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14 Apr 2014
TL;DR: The energy constraints of neuronal signaling within biology are examined, the quantitative tradeoff between energy use and information processing is reviewed, and whether the biophysics and design of nerve cells minimizes energy consumption is asked.
Abstract: Maintaining the ability of the nervous system to perceive, remember, process, and react to the outside world requires a continuous energy supply. Yet the overall power consumption is remarkably low, which has inspired engineers to mimic nervous systems in designing artificial cochlea, retinal implants, and brain-computer interfaces (BCIs) to improve the quality of life in patients. Such neuromorphic devices are both energy efficient and increasingly able to emulate many functions of the human nervous system. We examine the energy constraints of neuronal signaling within biology, review the quantitative tradeoff between energy use and information processing, and ask whether the biophysics and design of nerve cells minimizes energy consumption.
90 citations
Cites background from "Optimal Inputs for Phase Models of ..."
...One can ask what time course these synaptic currents should have to minimize energy consumption [69], [70], [84]....
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References
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TL;DR: This article concludes a series of papers concerned with the flow of electric current through the surface membrane of a giant nerve fibre by putting them into mathematical form and showing that they will account for conduction and excitation in quantitative terms.
Abstract: This article concludes a series of papers concerned with the flow of electric current through the surface membrane of a giant nerve fibre (Hodgkinet al, 1952,J Physiol116, 424–448; Hodgkin and Huxley, 1952,J Physiol116, 449–566) Its general object is to discuss the results of the preceding papers (Section 1), to put them into mathematical form (Section 2) and to show that they will account for conduction and excitation in quantitative terms (Sections 3–6)
19,800 citations
Book•
01 Jan 1969
5,681 citations
"Optimal Inputs for Phase Models of ..." refers background or methods in this paper
...(For higher dimensional neural models, such as the Hodgkin-Huxley equations considered below, gradient-based numerical models that iteratively update I(t) via the variational derivative δP δI(t) may be required; see [6]....
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...[6])....
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...in which the injected current takes the extreme values of ±Ī [6]....
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TL;DR: In this paper, the authors describe the rules of the ring, the ring population, and the need to get off the ring in order to measure the movement of a cyclic clock.
Abstract: 1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects
3,424 citations
Book•
01 Jan 1980
TL;DR: The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways is presented.
Abstract: 1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects
3,077 citations