A number of studies have suggested that many properties of brain activity can be understood in terms of critical systems. However it is still not known how the long-range susceptibilities characteristic of criticality arise in the living brain from its local connectivity structures. Here we prove that a dynamically critically-poised model of cortex acquires an infinitely-long ranged susceptibility in the absence of input. When an input is presented, the susceptibility attenuates exponentially as a function of distance, with an increasing spatial attenuation constant (i.e., decreasing range) the larger the input. This is in direct agreement with recent results that show that waves of local field potential activity evoked by single spikes in primary visual cortex of cat and macaque attenuate with a characteristic length that also increases with decreasing contrast of the visual stimulus. A susceptibility that changes spatial range with input strength can be thought to implement an input-dependent spatial integration: when the input is large, no additional evidence is needed in addition to the local input; when the input is weak, evidence needs to be integrated over a larger spatial domain to achieve a decision. Such input-strength-dependent strategies have been demonstrated in visual processing. Our results suggest that input-strength dependent spatial integration may be a natural feature of a critically-balanced cortical network.
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