Post by Zahra on Jun 14, 2016 6:29:57 GMT
To quantify the balanced networks in an irregular asynchronous state!
There is a section in [1] which explains about a method in which we can derive the PRC of a noisy oscillating system through its spiking pattern and the induced noise.
Then it links us to some references for example [2], in which Izhikevich superficially talk about the method he used for this purpose. In summary it is to minimize the square of the period error in the Taylor expansion of PRC, in order to optimized the PRC's Taylor coefficients.
The next step is the results that connect PRC to STA, for example in [3], they have demonstrated that the phase- resetting curve (PRC) from dynamics and the spike triggered average (STA) from statistical analysis, are closely related when neurons fire in a nearly regular manner and the stimulus is sufficiently small.
And next in [4] they have demonstrated that the phase response curve (PRC) can be reconstructed using a weighted spike- triggered average of an injected fluctuating input. The key idea is to choose the weight to be proportional to the magnitude of the fluctuation of the oscillatory period. Particularly, in this method we don't have the assumptions in [3], we can calculate the PRC in an irregular train.
[5]th analyzes the state-space structure of spiking neural networks, namely, sparse random networks of inhibitory leaky integrate-and-fire (LIF) neurons, in the balanced state (asynchronous irregular spike pattern). Current composed of constant excitatory external currents and inhibitory nondelayed pulses, finally the neurons driven by strong input fluctuations that result from a dynamical balance of excitatory and inhibitory inputs. Although capable of generating complex irregular spike sequences, they show that these networks actually exhibit negative-definite Lyapunov spectra. The spectra are invariant to the network size, hence this stable dynamics is extensive and preserved in the thermodynamic limit.
In this paper they do some calculations from voltage to phase and from transition curve to phase response curve of the neurons and then some stability investigations that are not so easy for me to follow.
From now on, I would not be so accurate and it is just for classifying the refs.
[6,7] look relevant, especially in [7] they try to define a parameter as STA of population rate, which is somehow strange for me!
SPIKE-TRIGGERED AVERAGE (STA) OF GLOBAL ACTIVITY. The spike triggered average is the cross-correlation between the spike train of a single neuron and the global activity. The instantaneous firing rate of a population during a 300-ms time window surrounding a spike (150 ms before and 150 ms after the spike time) is averaged over all spikes of the spike train of a single neuron.
There is some analytical works on phase response of a population:
the phase response of stochastic oscillators revealed the interplay between microscopic and macroscopic collective phase sensitivity, for ensembles of globally coupled [8], and network coupled elements [9]. In opposition to [8,9], [10] considers deterministic oscillators and perturbations of arbitrary strength (not infinitesimal), thus allowing the exploration of the phase reset’s time evolution.
[1] Destexhe, Alain, and Gif Yvette. 2009. Phase Response Curves in Neuroscience Theory, Experiment, and Analysis. Media. Vol. 8. doi:10.1007/978-0-387-93797-7.
[2] Izhikevich, Eugene M. 2007. Dynamical Systems in Neuroscience Computational Neuroscience. Dynamical Systems. Vol. 25. doi:10.1017/S0143385704000173.
[3] Ermentrout, G. Bard, Roberto F. Galan, and Nathaniel N. Urban. 2007. “Relating Neural Dynamics to Neural Coding.” Physical Review Letters 99 (24): 1–4. doi:10.1103/PhysRevLett.99.248103.
[4] Ota, Kaiichiro, Masaki Nomura, and Toshio Aoyagi. 2009. “Weighted Spike-Triggered Average of a Fluctuating Stimulus Yielding the Phase Response Curve.” Physical Review Letters 103 (2). doi:10.1103/PhysRevLett.103.024101.
[5] Monteforte, Michael, and Fred Wolf. 2012. “Dynamic Flux Tubes Form Reservoirs of Stability in Neuronal Circuits.” Physical Review X 2 (4): 1–12. doi:10.1103/PhysRevX.2.041007.
[6] Brunel, Nicolas, and Xiao-Jing Wang. 2003. “What Determines the Frequency of Fast Network Oscillations with Irregular Neural Discharges? I. Synaptic Dynamics and Excitation-Inhibition Balance.” Journal of Neurophysiology 90 (1): 415–30. doi:10.1152/jn.01095.2002.
[7] Geisler, Caroline, Nicolas Brunel, and Xiao-Jing Wang. 2005. “Contributions of Intrinsic Membrane Dynamics to Fast Network Oscillations with Irregular Neuronal Discharges.” J Neurophysiol 94 (6): 4344–61. doi:10.1152/jn.00510.2004.
[8] Kawamura, Yoji, Hiroya Nakao, Kensuke Arai, Hiroshi Kori, and Yoshiki Kuramoto. 2008. “Collective Phase Sensitivity.” Physical Review Letters 101 (2): 1–4. doi:10.1103/PhysRevLett.101.024101.
[9] Kori, Hiroshi, Yoji Kawamura, Hiroya Nakao, Kensuke Arai, and Yoshiki Kuramoto. 2009. “Collective-Phase Description of Coupled Oscillators with General Network Structure.” Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 80 (3). doi:10.1103/PhysRevE.80.036207.
[10] Levnajić, Zoran, and Arkady Pikovsky. 2010. “Phase Resetting of Collective Rhythm in Ensembles of Oscillators.” Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 82 (5): 1–10. doi:10.1103/PhysRevE.82.056202.
There is a section in [1] which explains about a method in which we can derive the PRC of a noisy oscillating system through its spiking pattern and the induced noise.
Then it links us to some references for example [2], in which Izhikevich superficially talk about the method he used for this purpose. In summary it is to minimize the square of the period error in the Taylor expansion of PRC, in order to optimized the PRC's Taylor coefficients.
The next step is the results that connect PRC to STA, for example in [3], they have demonstrated that the phase- resetting curve (PRC) from dynamics and the spike triggered average (STA) from statistical analysis, are closely related when neurons fire in a nearly regular manner and the stimulus is sufficiently small.
And next in [4] they have demonstrated that the phase response curve (PRC) can be reconstructed using a weighted spike- triggered average of an injected fluctuating input. The key idea is to choose the weight to be proportional to the magnitude of the fluctuation of the oscillatory period. Particularly, in this method we don't have the assumptions in [3], we can calculate the PRC in an irregular train.
[5]th analyzes the state-space structure of spiking neural networks, namely, sparse random networks of inhibitory leaky integrate-and-fire (LIF) neurons, in the balanced state (asynchronous irregular spike pattern). Current composed of constant excitatory external currents and inhibitory nondelayed pulses, finally the neurons driven by strong input fluctuations that result from a dynamical balance of excitatory and inhibitory inputs. Although capable of generating complex irregular spike sequences, they show that these networks actually exhibit negative-definite Lyapunov spectra. The spectra are invariant to the network size, hence this stable dynamics is extensive and preserved in the thermodynamic limit.
In this paper they do some calculations from voltage to phase and from transition curve to phase response curve of the neurons and then some stability investigations that are not so easy for me to follow.
From now on, I would not be so accurate and it is just for classifying the refs.
[6,7] look relevant, especially in [7] they try to define a parameter as STA of population rate, which is somehow strange for me!
SPIKE-TRIGGERED AVERAGE (STA) OF GLOBAL ACTIVITY. The spike triggered average is the cross-correlation between the spike train of a single neuron and the global activity. The instantaneous firing rate of a population during a 300-ms time window surrounding a spike (150 ms before and 150 ms after the spike time) is averaged over all spikes of the spike train of a single neuron.
There is some analytical works on phase response of a population:
the phase response of stochastic oscillators revealed the interplay between microscopic and macroscopic collective phase sensitivity, for ensembles of globally coupled [8], and network coupled elements [9]. In opposition to [8,9], [10] considers deterministic oscillators and perturbations of arbitrary strength (not infinitesimal), thus allowing the exploration of the phase reset’s time evolution.
[1] Destexhe, Alain, and Gif Yvette. 2009. Phase Response Curves in Neuroscience Theory, Experiment, and Analysis. Media. Vol. 8. doi:10.1007/978-0-387-93797-7.
[2] Izhikevich, Eugene M. 2007. Dynamical Systems in Neuroscience Computational Neuroscience. Dynamical Systems. Vol. 25. doi:10.1017/S0143385704000173.
[3] Ermentrout, G. Bard, Roberto F. Galan, and Nathaniel N. Urban. 2007. “Relating Neural Dynamics to Neural Coding.” Physical Review Letters 99 (24): 1–4. doi:10.1103/PhysRevLett.99.248103.
[4] Ota, Kaiichiro, Masaki Nomura, and Toshio Aoyagi. 2009. “Weighted Spike-Triggered Average of a Fluctuating Stimulus Yielding the Phase Response Curve.” Physical Review Letters 103 (2). doi:10.1103/PhysRevLett.103.024101.
[5] Monteforte, Michael, and Fred Wolf. 2012. “Dynamic Flux Tubes Form Reservoirs of Stability in Neuronal Circuits.” Physical Review X 2 (4): 1–12. doi:10.1103/PhysRevX.2.041007.
[6] Brunel, Nicolas, and Xiao-Jing Wang. 2003. “What Determines the Frequency of Fast Network Oscillations with Irregular Neural Discharges? I. Synaptic Dynamics and Excitation-Inhibition Balance.” Journal of Neurophysiology 90 (1): 415–30. doi:10.1152/jn.01095.2002.
[7] Geisler, Caroline, Nicolas Brunel, and Xiao-Jing Wang. 2005. “Contributions of Intrinsic Membrane Dynamics to Fast Network Oscillations with Irregular Neuronal Discharges.” J Neurophysiol 94 (6): 4344–61. doi:10.1152/jn.00510.2004.
[8] Kawamura, Yoji, Hiroya Nakao, Kensuke Arai, Hiroshi Kori, and Yoshiki Kuramoto. 2008. “Collective Phase Sensitivity.” Physical Review Letters 101 (2): 1–4. doi:10.1103/PhysRevLett.101.024101.
[9] Kori, Hiroshi, Yoji Kawamura, Hiroya Nakao, Kensuke Arai, and Yoshiki Kuramoto. 2009. “Collective-Phase Description of Coupled Oscillators with General Network Structure.” Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 80 (3). doi:10.1103/PhysRevE.80.036207.
[10] Levnajić, Zoran, and Arkady Pikovsky. 2010. “Phase Resetting of Collective Rhythm in Ensembles of Oscillators.” Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 82 (5): 1–10. doi:10.1103/PhysRevE.82.056202.
