Stability of bumps in a two-population neural-field model with quasi-power temporal kernels

In the present work, we investigate the annihilation of persistent localized activity states (bumps) in a Wilson-Cowan type two-population neural field model in response to α-type spatio-temporal external input. These activity states serves as working memory in the prefrontal cortex. The impact of different parameters involved in the external input on annihilation of these persistent activity states is investigated in detail. The α-type temporal function in the external input is closer to natural phenomenon as observed in Roth et. al. (Nature Neuroscience, vol. 19 (2016), 229-307). Two types of eraser mechanism are used in this work to annihilate the spatially symmetric solutions. Initially, if there is an activity in the network, inhibitory external input with no excitatory part and over excitation with no inhibition in the external input can kill the activity. Our results show that the annihilation of persistent activity states using α-type temporal function in the external input is more roubust and more efficient as compare to triangular one as used by Yousaf et al. (Neural networks., vol. 46 (2013), pp. 75-90). It is also found that the relative inhibition time constant plays a crucial role in annihilation of the activity. Runge-Kutta fourth order method has been employed for numerical simulations of this work. INDEX TERMS Annihilation of bumps, αfunction, integro-differential equations, neural networks, Runge-Kutta fourth order method, symmetric solutions.

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“…This impact decays exponentialy with time. The common choice for the functions α m are [2], [17], [21], [22]:…”

Section: Modelmentioning

confidence: 99%

“…The system of Volterra equations (2) with exponentially decaying temporal kernals is transformed into the following integro-differential equations [17], [21], [23], [24] ∂v…”

Section: Modelmentioning

confidence: 99%

Effect of Alpha-Type External Input on Annihilation of Self-Sustained Activity in a Two Population Neural Field Model

et al. 2019

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“…attractor states in the network, persistent activation of thalmo-cortical and cortiocortical loops [12], [13]. Especially, the idea of network attractor states in framework of neural field models investigated intensively in many studies (e.g., [3], [4], [6], [14], [16], [18], [20], [25], [27]) Working memory are generally discussed the disjoint classes on how the persistent states of activity is generated. One most popular mechanism, in the cell assembly the activity is persistent through strong recurrent excitatory connections [21].…”

Section: Introductionmentioning

confidence: 99%

Emergence of Persistent Activity States in a Two-Population Neural Field Model for Smooth $\alpha$ -Type External Input

et al. 2019

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We investigate the emergence of localized activity states, so-called bumps in Wilson-Cowan type two-population neural field model under the influence of transient spatio-temporal external input with smooth α-type temporal function. This two-population model is composed of two coupled nonlinear differential equations derived for the dynamics of spatially localized populations of both excitatory and inhibitory model neurons. The model with no external input corresponds to at most two bump pair solutions. Such a system can be interpreted as a minimal cortical model for short term working memory, that is the ability of the brain to actively hold stimulus-related information for some seconds in short term memory and discards once it becomes irrelevant Initially, if there is no activity in the system, persistent activity state can be evoked by switching on a suitable transient excitatory external input. This activity remains stable even though external input is switched off. The effect of external input on the emergence of bumps for different spatial and smooth α-type temporal functions of external input is investigated and found that certain parameters play a key role in the generation of persistent activity states in the network, e.g., relative inhibition time constant, total duration, and the amplitude of external input. It is found that the minimum values of the amplitude and active time to evoke the activity in the network is smaller than those observed in Yousaf et al. showing that the present choice of temporal function in the external input is more effective and more close to natural behavior. INDEX TERMS Two population neuronal networks, stationary symmetric solutions, integro-differential equations, Runge-Kutta Fourth order method.

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“…In recurrent networks such localized stationary states are naturally formed by a combination of (i) a strong and localized recurrent excitation boosting the bump and (ii) a spatially more extended 'lateral' inhibition preventing the bump from growing in size [9]. Neuronal field models have provided a powerful and versatile tool for the investigation of the properties of such bump states [8,9,10,11,12,13], and a large number of studies have used such models to study generic properties of bumps such as conditions for their existence and stability [14,15,16,17,18,19,20,21,22,23,24,25,26,27].…”

Section: Introductionmentioning

confidence: 99%

“…In the present study we investigate the effects of spatially localized external inputs on bump states in a two-population Wilson-Cowan like model with one excitatory and one inhibitory population. We have previously investigated bump states in this model without external inputs [24,27], and an interesting feature is the key role played by the inhibitory time constant in determining the stability of bumps. The bumps are found to be stable only for inhibitory time constants below a critical value, about three times the excitatory time constant for the example in [24], while the bumps are converted to stable breathers through a Hopf bifurcation at the critical value.…”

Section: Introductionmentioning

confidence: 99%

Effect of localized input on bump solutions in a two-population neural-field model

Yousaf¹,

Wyller²,

Tetzlaff³

et al. 2013

Nonlinear Analysis: Real World Applications

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We investigate a two-population Wilson-Cowan model extended with stationary and spatially dependent localized external inputs and study the existence and stability of localized stationary (bump) solutions. The generic situation for this model in the absence of external inputs corresponds to two bump pairs, one narrow and one broad pair. For spatially wide localized external inputs we find this generic picture to be unchanged. However, for strongly localized external inputs we find that three and even four bump pairs, all with symmetric activity profiles around the center of the localized external input, may coexist. We next investigate the stability of these bump pairs using two different techniques: a simplified phase-space reduction (Amari) technique and full stability analysis. Examples of models, i.e., choices of model parameters, exhibiting up to three stable bump pairs are found. The Amari technique is further found to be a poor predictor of stability in the case of strongly localized external inputs. The bump-pair states are also probed numerically using a fourth order Runge-Kutta method, and an excellent agreement is found between numerical simulations and the analytical predictions from full stability analysis.

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Stability of bumps in a two-population neural-field model with quasi-power temporal kernels

Cited by 5 publications

References 68 publications

Effect of Alpha-Type External Input on Annihilation of Self-Sustained Activity in a Two Population Neural Field Model

Effect of Alpha-Type External Input on Annihilation of Self-Sustained Activity in a Two Population Neural Field Model

Emergence of Persistent Activity States in a Two-Population Neural Field Model for Smooth $\alpha$ -Type External Input

Effect of localized input on bump solutions in a two-population neural-field model

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