Stimulus-Driven Traveling Solutions in Continuum Neuronal Models with a General Smooth Firing Rate Function

Ermentrout, Bard; Jalics, Jozsi Z.; Rubin, Jonathan E.

doi:10.1137/090775737

Cited by 43 publications

(39 citation statements)

References 25 publications

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“…This problem has been studied in 1D using both the constructive method for Heaviside nonlinearities [112] and singular perturbation methods for smooth F [147]. We discuss the former approach here.…”

Section: Stimulus-driven Bumpsmentioning

confidence: 99%

Spatiotemporal dynamics of continuum neural fields

Bressloff¹

2011

J. Phys. A: Math. Theor.

362

441

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We survey recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields. Neural fields model the large-scale dynamics of spatially structured biological neural networks in terms of nonlinear integrodifferential equations whose associated integral kernels represent the spatial distribution of neuronal synaptic connections. They provide an important example of spatially extended excitable systems with nonlocal interactions and exhibit a wide range of spatially coherent dynamics including traveling waves oscillations and Turing-like patterns.

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Section: Stimulus-driven Bumpsmentioning

confidence: 99%

Spatiotemporal dynamics of continuum neural fields

Bressloff¹

2011

J. Phys. A: Math. Theor.

362

441

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“…In particular, it would be interesting to see how a negative feedback variable like a separate inhibitory population [14], spike frequency adaptation [21,29], or synaptic depression [26] would affect the shape of the response function. Adjoints for models with linear recovery have been calculated previously in [32,51]. Type II response functions in lateral inhibitory networks may be separable into positive and negative parts in analogous two population networks, depending on whether the excitatory or inhibitory population is stimulated.…”

Section: Discussionmentioning

confidence: 99%

“…For moving inputs, stimulus-locked traveling waves can undergo a Hopf bifurcation beyond which traveling breathers exist [31,32]. Pulsating fronts can arise when a spatially periodic stationary input is applied [33].…”

Section: Introductionmentioning

confidence: 99%

Response of traveling waves to transient inputs in neural fields

Kilpatrick

Ermentrout

2012

Phys. Rev. E

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We analyze the effects of transient stimulation on traveling waves in neural field equations. Neural fields are modeled as integrodifferential equations whose convolution term represents the synaptic connections of a spatially extended neuronal network. The adjoint of the linearized wave equation can be used to identify how a particular input will shift the location of a traveling wave. This wave response function is analogous to the phase response curve of limit cycle oscillators. For traveling fronts in an excitatory network, the sign of the shift depends solely on the sign of the transient input. A complementary estimate of the effective shift is derived using an equation for the timedependent speed of the perturbed front. Traveling pulses are analyzed in an asymmetric lateral inhibitory network, and they can be advanced or delayed, depending on the position of spatially localized transient inputs. We also develop bounds on the amplitude of transient input necessary to terminate traveling pulses, based on the global bifurcation structure of the neural field.

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“…4.8. This analysis for stimulus-locked fronts was carried out in [6] and an extension of stimulus-locked bumps for a general smooth firing rate function F was studied in [20]. The essential spectrum lies within the set D D fz W Re z 2 OE Re C ; Re g, where Re ˙> 0, inducing no instability [25,41,55].…”

Section: Natural and Stimulus-locked Traveling Activity Bumpsmentioning

confidence: 99%

Spatiotemporal Pattern Formation in Neural Fields with Linear Adaptation

2014

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We study spatiotemporal patterns of activity that emerge in neural fields in the presence of linear adaptation. Using an amplitude equation approach, we show that bifurcations from the homogeneous rest state can lead to a wide variety of stationary and propagating patterns on one-and two-dimensional periodic domains, particularly in the case of lateral-inhibitory synaptic weights. Other typical solutions are stationary and traveling localized activity bumps; however, we observe exotic time-periodic localized patterns as well. Using linear stability analysis that perturbs about stationary and traveling bump solutions, we study conditions for the activity to lock to a stationary or traveling external input on both periodic and infinite one-dimensional spatial domains. Hopf and saddle-node bifurcations can signify the boundary beyond which stationary or traveling bumps fail to lock to external inputs. Just beyond a Hopf bifurcation point, activity bumps often begin to oscillate, becoming breather or slosher solutions.

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Stimulus-Driven Traveling Solutions in Continuum Neuronal Models with a General Smooth Firing Rate Function

Cited by 43 publications

References 25 publications

Spatiotemporal dynamics of continuum neural fields

Spatiotemporal dynamics of continuum neural fields

Response of traveling waves to transient inputs in neural fields

Spatiotemporal Pattern Formation in Neural Fields with Linear Adaptation

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