Synchrony in excitatory neural networks
Neural Computation
Weakly connected neural networks
Weakly connected neural networks
Elements of applied bifurcation theory (2nd ed.)
Elements of applied bifurcation theory (2nd ed.)
Simulating, Analyzing, and Animating Dynamical Systems: A Guide Toi Xppaut for Researchers and Students
MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs
ACM Transactions on Mathematical Software (TOMS)
Numerical Periodic Normalization for Codim 1 Bifurcations of Limit Cycles
SIAM Journal on Numerical Analysis
Type i membranes, phase resetting curves, and synchrony
Neural Computation
Proceedings of the 2009 International Conference on Computer-Aided Design
Population encoding with Hodgkin-Huxley neurons
IEEE Transactions on Information Theory - Special issue on information theory in molecular biology and neuroscience
Convergence analysis of a numerical method to solve the adjoint linearized periodic orbit equations
Applied Numerical Mathematics
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
Phase response curves, delays and synchronization in MATLAB
ICCS'06 Proceedings of the 6th international conference on Computational Science - Volume Part II
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Neurons are often modeled by dynamical systems—parameterized systems of differential equations. A typical behavioral pattern of neurons is periodic spiking; this corresponds to the presence of stable limit cycles in the dynamical systems model. The phase resetting and phase response curves (PRCs) describe the reaction of the spiking neuron to an input pulse at each point of the cycle. We develop a new method for computing these curves as a by-product of the solution of the boundary value problem for the stable limit cycle. The method is mathematically equivalent to the adjoint method, but our implementation is computationally much faster and more robust than any existing method. In fact, it can compute PRCs even where the limit cycle can hardly be found by time integration, for example, because it is close to another stable limit cycle. In addition, we obtain the discretized phase response curve in a form that is ideally suited for most applications. We present several examples and provide the implementation in a freely available Matlab code.