Macroscopic phase-resetting curves for spiking neural networks

Dumont, Grégory; Ermentrout, G. Bard; Gutkin, Boris

doi:10.1103/physreve.96.042311

Cited by 39 publications

(53 citation statements)

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“…In this paper, we introduced variability through external noise much as earlier work by Brunel et al have done in the leaky integrate and fire model [13]. Recently [14,36,38] have explored the macroscopic dynamics of spiking networks of neurons that have quenched variability; that is, rather than external noise as in the present paper, they have introduced randomness in the parameters, mainly the applied currents to the neurons. By using the quadratic integrate and fire (or, equivalently, the theta neuron), they are able to use the Ott-Antonsen ansatz [37], to reduce the FPE to a low-dimensional dynamical system as long as the random parameters are taken from a Lorentzian distribution.…”

Section: Discussionmentioning

confidence: 97%

“…For example, with instantaneous synaptic connections, an EI network becomes a simple four-dimensional dynamical system that produces PING type oscillations. [38] used this method to compute the phase response functions of such an EI network and compared the results to the full spiking simulations. The techniques used in [38] should be applicable to the present model albeit with quenched variability.…”

Section: Discussionmentioning

confidence: 99%

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Relationship between the mechanisms of gamma rhythm generation and the magnitude of the macroscopic phase response function in a population of excitatory and inhibitory modified quadratic integrate-and-fire neurons

et al. 2018

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Gamma oscillations are thought to play an important role in brain function. Interneuron gamma (ING) and pyramidal interneuron gamma (PING) mechanisms have been proposed as generation mechanisms for these oscillations. However, the relation between the generation mechanisms and the dynamical properties of the gamma oscillation are still unclear. Among the dynamical properties of the gamma oscillation, the phase response function (PRF) is important because it encodes the response of the oscillation to inputs. Recently, the PRF for an inhibitory population of modified theta neurons that generate an ING rhythm was computed by the adjoint method applied to the associated Fokker-Planck equation (FPE) for the model. The modified theta model incorporates conductance-based synapses as well as the voltage and current dynamics. Here, we extended this previous work by creating an excitatory-inhibitory (E-I) network using the modified theta model and described the population dynamics with the corresponding FPE. We conducted a bifurcation analysis of the FPE to find parameter regions which generate gamma oscillations. In order to label the oscillatory parameter regions by their generation mechanisms, we defined ING- and PING-type gamma oscillation in a mathematically plausible way based on the driver of the inhibitory population. We labeled the oscillatory parameter regions by these generation mechanisms and derived PRFs via the adjoint method on the FPE in order to investigate the differences in the responses of each type of oscillation to inputs. PRFs for PING and ING mechanisms are derived and compared. We found the amplitude of the PRF for the excitatory population is larger in the PING case than in the ING case. Finally, the E-I population of the modified theta neuron enabled us to analyze the PRFs of PING-type gamma oscillation and the entrainment ability of E and I populations. We found a parameter region in which PRFs of E and I are both purely positive in the case of PING oscillations. The different entrainment abilities of E and I stimulation as governed by the respective PRFs was compared to direct simulations of finite populations of model neurons. We find that it is easier to entrain the gamma rhythm by stimulating the inhibitory population than by stimulating the excitatory population as has been found experimentally.

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Section: Discussionmentioning

confidence: 97%

Section: Discussionmentioning

confidence: 99%

Relationship between the mechanisms of gamma rhythm generation and the magnitude of the macroscopic phase response function in a population of excitatory and inhibitory modified quadratic integrate-and-fire neurons

et al. 2018

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“…October 16, 2018 28/34curves of the population dynamics[43] to the data presented here. Furthermore, the…”

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confidence: 83%

Predicting the effects of deep brain stimulation using a reduced coupled oscillator model

Weerasinghe

Duchet

Cagnan

et al. 2018

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Deep brain stimulation (DBS) is known to be an effective treatment for a variety of neurological disorders, including Parkinson's disease and essential tremor (ET). At present, it involves administering a train of pulses with constant frequency via electrodes implanted into the brain. New 'closed-loop' approaches involve delivering stimulation according to the ongoing symptoms or brain activity and have the potential to provide improvements in terms of efficiency, efficacy and reduction of side effects.The success of closed-loop DBS depends on being able to devise a stimulation strategy that minimizes oscillations in neural activity associated with symptoms of motor disorders. A useful stepping stone towards this is to construct a mathematical model, which can describe how the brain oscillations should change when stimulation is applied at a particular state of the system. Our work focuses on the use of coupled oscillators to represent neurons in areas generating pathological oscillations. Using a reduced form of the Kuramoto model, we analyse how a patient should respond to stimulation when Introduction 1 Symptoms of several neurological disorders are thought to arise from overly synchronous 2 activity within neural populations. The severity of clinical impairment in Parkinson's 3 disease (PD) is known to be correlated with an increase in the beta (13-35 Hz) 4 oscillations in the local field potential (LFP) and in the activity of individual neurons in 5October 16, 2018 2/34 the basal ganglia [1, 2]. The tremor symptoms associated with essential tremor (ET) are 6 thought to arise from synchronous activity in a network of brain areas including the 7 thalamus [3]. In both PD and ET the muscle activity driving the tremor is coherent 8 with local field potentials in the thalamus [4] and bursts of spikes produced by 9 individual thalamic neurons [5]. 10 Deep brain stimulation (DBS) is a well-established treatment option for PD and ET 11 which involves delivering stimulation via electrodes implanted into the brain. The 12 present generation of the technology involves manually tuning the parameters of 13 stimulation, such as the pulse width, frequency and intensity in an attempt to achieve 14 the best treatment. In particular, the choice of frequency is known to be crucial for 15 efficacy, and high frequency DBS (120-180 Hz) has been found to be effective for PD 16 and ET patients [6]. High frequency DBS is known to suppress the pathological 17 oscillations occurring in PD [7], but despite its long history, the underlying mechanisms 18 causing this suppression remain unclear, and several distinct theories have been 19 proposed [8-10]. One influential theory suggests that high frequency DBS activates 20 target neurons to such an extent that their synaptic transmission becomes saturated 21 and they are no longer able to transmit pathological oscillations [11]. Since high 22 frequency DBS can cause side-effects such as speech-impairments [12] and gambling 23 tendencies [13], improvements to this treatment approach are desi...

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“…where the matrix L is given by Eq. (41). The forcing of the nth module F f orc n can be decomposed as a sum of inputs coming from the mean activities, I (0) (t + φ m ), of nearby modules and of fluctuations of their activities,…”

Section: Stochastic Dynamics Of the Phases Of Oscillations Along A Chmentioning

confidence: 99%

“…An exception is the soluble case of deterministic quadratic integrate-and-fire neurons with a wide distribution of frequencies [40] which has been used to study oscillations of an E-I module [41] as well as synchonization between two weakly-coupled modules [42]. Aside from oscillations, the conditions necessary for the existence of a balanced state [43] in a spatially structured network have been studied [44] and it was found in particular that the spatial spread of excitation should be broader than that of inhibitory inputs.…”

Section: Introductionmentioning

confidence: 99%

Synchronization, stochasticity and phase waves in neuronal networks with spatially-structured connectivity

Kulkarni

Ranft

Hakim

2020

Preprint

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These authors contributed equally to this work. AbstractOscillations in the beta/low gamma range (10-45 Hz) are recorded in diverse neural structures. They have successfully been modeled as sparsely synchronized oscillations arising from reciprocal interactions between randomly connected excitatory (E) pyramidal cells and local interneurons (I). The synchronization of spatially distant oscillatory spiking E-I modules has been well studied in the rate model framework but less so for modules of spiking neurons. Here, we first show that previously proposed modifications of rate models provide a quantitative description of spiking E-I modules of Exponential Integrate-and-Fire (EIF) neurons. This allows us to analyze the dynamical regimes of sparsely synchronized oscillatory E-I modules connected by long-range excitatory interactions, for two modules, as well as for a chain of such modules. For modules with a large number of neurons (> 10 5 ), we obtain results similar to previously obtained ones based on the classic deterministic Wilson-Cowan rate model, with the added bonus that the results quantitatively describe simulations of spiking EIF neurons. However, for modules with a moderate (∼ 10 4 ) number of neurons, stochastic variations in the spike emission of neurons are important and need to be taken into account. On the one hand, they modify the oscillations in a way that tends to promote synchronization between different modules. On the other hand, independent fluctuations on different modules tend to disrupt synchronization. The correlations between distant oscillatory modules can be described by stochastic equations for the oscillator phases that have been intensely studied in other contexts. On shorter distances, we develop a description that also takes into account amplitude modes and that quantitatively accounts for our simulation data. Stochastic dephasing of neighboring modules produces transient phase gradients and the transient appearance of phase waves. We propose that these stochastically-induced phase waves provide an explanative framework for the observations of traveling waves in the cortex during beta oscillations.

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Macroscopic phase-resetting curves for spiking neural networks

Cited by 39 publications

References 53 publications

Relationship between the mechanisms of gamma rhythm generation and the magnitude of the macroscopic phase response function in a population of excitatory and inhibitory modified quadratic integrate-and-fire neurons

Relationship between the mechanisms of gamma rhythm generation and the magnitude of the macroscopic phase response function in a population of excitatory and inhibitory modified quadratic integrate-and-fire neurons

Predicting the effects of deep brain stimulation using a reduced coupled oscillator model

Synchronization, stochasticity and phase waves in neuronal networks with spatially-structured connectivity

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