Single bumps in a 2-population homogenized neuronal network model

Kolodina, Karina; Oleynik, Anna; Wyller, John

doi:10.1016/j.physd.2018.01.004

Cited by 2 publications

(10 citation statements)

References 31 publications

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“…The numerical scheme is realized in MATLAB © with the build-in functions conv2 and ode45. Just as in Wyller et al [15] for the translational invariant case (1), we detect numerically in the steep firing rate regime and in the weakly heterogeneous case that the final stage of the pattern forming process consists of stable spatial oscillations where the shape of each oscillation matches remarkably well with the shape of the 1-bumps found in Kolodina et al [33]. We conjecture that these oscillations are y-independent, 1-bump periodic solutions of (4) in the steep firing rate regime.…”

Section: Introductionsupporting

confidence: 87%

“…The present investigation complements the papers [14,15], and [33] as well as the works [25,30,31], and [24].…”

Section: Introductionsupporting

confidence: 64%

“…We are not aware of any studies of the microstructure effects on the pattern formation mechanism within such modeling frameworks either. In Kolodina et al [33] the existence and stability of y-independent single bumps in the homogenized 2-population model (4) in one spatial dimension (N = 1) were investigated, with the firing rate functions modeled by means of the Heaviside function.…”

Section: Introductionmentioning

confidence: 99%

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Pattern formation in a 2-population homogenized neuronal network model

et al. 2021

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We study pattern formation in a 2-population homogenized neural field model of the Hopfield type in one spatial dimension with periodic microstructure. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. It is shown that the nonlocal synaptic interactions promote a finite band width instability. The stability method relies on a sequence of wave-number dependent invariants of $2\times 2$ 2 × 2 -stability matrices representing the sequence of Fourier-transformed linearized evolution equations for the perturbation imposed on the homogeneous background. The generic picture of the instability structure consists of a finite set of well-separated gain bands. In the shallow firing rate regime the nonlinear development of the instability is determined by means of the translational invariant model with connectivity kernels replaced with the corresponding period averaged connectivity functions. In the steep firing rate regime the pattern formation process depends sensitively on the spatial localization of the connectivity kernels: For strongly localized kernels this process is determined by the translational invariant model with period averaged connectivity kernels, whereas in the complementary regime of weak and moderate localization requires the homogenized model as a starting point for the analysis. We follow the development of the instability numerically into the nonlinear regime for both steep and shallow firing rate functions when the connectivity kernels are modeled by means of an exponentially decaying function. We also study the pattern forming process numerically as a function of the heterogeneity parameters in four different regimes ranging from the weakly modulated case to the strongly heterogeneous case. For the weakly modulated regime, we observe that stable spatial oscillations are formed in the steep firing rate regime, whereas we get spatiotemporal oscillations in the shallow regime of the firing rate functions.

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Section: Introductionsupporting

confidence: 87%

“…The present investigation complements the papers [14,15], and [33] as well as the works [25,30,31], and [24].…”

Section: Introductionsupporting

confidence: 64%

Section: Introductionmentioning

confidence: 99%

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Pattern formation in a 2-population homogenized neuronal network model

et al. 2021

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“…The operator ⊗ in (1) defines the spatial convolution integral, given as (6) and the temporal convolution integral α mn * f is given as…”

Section: Modelmentioning

confidence: 99%

“…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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Single bumps in a 2-population homogenized neuronal network model

Cited by 2 publications

References 31 publications

Pattern formation in a 2-population homogenized neuronal network model

Pattern formation in a 2-population homogenized neuronal network model

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

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