Mathematical Equivalence of Two Common Forms of Firing Rate Models of Neural Networks

Miller, Kenneth D.; Fumarola, Francesco

doi:10.1162/neco_a_00221

Cited by 48 publications

(49 citation statements)

References 13 publications

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“…A decision is recorded when the first activity x j (t) exceeds a fixed threshold x j,th , adjustment of which is used to tune the speed-accuracy trade-off. Similar effects may be obtained by changing baseline activity or initial conditions, mechanisms that also appear likely, and to which we Grossberg (1988), Rumelhart and McClelland (1986) and Usher and McClelland (2001) for background on related connectionist networks, and Miller and Fumarola (2012) on the equivalence of different integrator models. The function f(Á) characterizing neural response is typically sigmoidal:…”

Section: Leaky Competing Accumulators and Drift-diffusion Processessupporting

confidence: 65%

“…Similar effects may be obtained by changing baseline activity or initial conditions, mechanisms that also appear likely, and to which we return in Section . See Grossberg (), Rumelhart and McClelland () and Usher and McClelland () for background on related connectionist networks, and Miller and Fumarola () on the equivalence of different integrator models.…”

Section: Optimal Performance In Simple Decisionsmentioning

confidence: 99%

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Optimality and Some of Its Discontents: Successes and Shortcomings of Existing Models for Binary Decisions

Holmes

Cohen

2014

Topics in Cognitive Science

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We review how leaky competing accumulators (LCAs) can be used to model decision making in two-alternative, forced-choice tasks, and we show how they reduce to drift diffusion (DD) processes in special cases. As continuum limits of the sequential probability ratio test, DD processes are optimal in producing decisions of specified accuracy in the shortest possible time. Furthermore, the DD model can be used to derive a speed-accuracy trade-off that optimizes reward rate for a restricted class of two alternative forced-choice decision tasks. We review findings that compare human performance with this benchmark, and we reveal both approximations to and deviations from optimality. We then discuss three potential sources of deviations from optimality at the psychological level-avoidance of errors, poor time estimation, and minimization of the cost of control-and review recent theoretical and empirical findings that address these possibilities. We also discuss the role of cognitive control in changing environments and in modulating exploitation and exploration. Finally, we consider physiological factors in which nonlinear dynamics may also contribute to deviations from optimality.

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Section: Leaky Competing Accumulators and Drift-diffusion Processessupporting

confidence: 65%

Section: Optimal Performance In Simple Decisionsmentioning

confidence: 99%

Optimality and Some of Its Discontents: Successes and Shortcomings of Existing Models for Binary Decisions

Holmes

Cohen

2014

Topics in Cognitive Science

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“…At least in the case where all neurons have equal time constants, i.e. T ∝ 1 , the two formulations are equivalent and are related by the change of variable v = W r + I r [46]…”

mentioning

confidence: 99%

Properties of networks with partially structured and partially random connectivity

Fumarola²,

2015

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Networks studied in many disciplines, including neuroscience and mathematical biology, have connectivity that may be stochastic about some underlying mean connectivity represented by a nonnormal matrix. Furthermore the stochasticity may not be i.i.d. across elements of the connectivity matrix. More generally, the problem of understanding the behavior of stochastic matrices with nontrivial mean structure and correlations arises in many settings. We address this by characterizing large random N × N matrices of the form A = M + LJR, where M, L and R are arbitrary deterministic matrices and J is a random matrix of zero-mean independent and identically distributed elements. M can be nonnormal, and L and R allow correlations that have separable dependence on row and column indices. We first provide a general formula for the eigenvalue density of A. For A nonnormal, the eigenvalues do not suffice to specify the dynamics induced by A, so we also provide general formulae for the transient evolution of the magnitude of activity and frequency power spectrum in an N -dimensional linear dynamical system with a coupling matrix given by A. These quantities can also be thought of as characterizing the stability and the magnitude of the linear response of a nonlinear network to small perturbations about a fixed point. We derive these formulae and work them out analytically for some examples of M, L and R motivated by neurobiological models. We also argue that the persistence as N → ∞ of a finite number of randomly distributed outlying eigenvalues outside the support of the eigenvalue density of A, as previously observed, arises in regions of the complex plane Ω where there are nonzero singular values of L−1(z1 − M)R−1 (for z ∈ Ω) that vanish as N → ∞. When such singular values do not exist and L and R are equal to the identity, there is a correspondence in the normalized Frobenius norm (but not in the operator norm) between the support of the spectrum of A for J of norm σ and the σ-pseudospectrum of M.

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“…The first term in the right-hand side of Equation 1 represents the leak current; this type of firing rate equation corresponds to the subthreshold dynamics of leaky integrate-and-fire neurons (Miller and Fumarola, 2012). The selected default parameters are Fmax=20, β=1, θ=0.5, r=1 and E=0 unless stated otherwise.…”

Section: Recurrent Network Are Unstable Without Sensory Evidencementioning

confidence: 99%

Cortical circuit-based lossless neural integrator for perceptual decision-making

Lee

Tsunada

Vijayan

et al. 2017

Preprint

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SummaryThe intrinsic uncertainty of sensory information (i.e., evidence) does not necessarily deter an observer from making a reliable decision. It is believed that uncertainty is reduced by integrating (accumulating) incoming sensory evidence. Traditionally, recurrent network models have been thought to underlie the integration of evidence necessary for decision-making, but they have yet to fully explain humans' ability to hold accumulated evidence during temporal gaps in sensory inputs. In our study, using a computational model, we show that cortical circuits can readily explain humans' ability to make robust decisions during temporal gaps. Specifically, our model predicts that the interplay among depressing synapses and the two major interneuron types allow the cortex to switch flexibly between 'retention' and 'integration' modes. The evidence can be read out in two different modes mirroring 'stepping' and 'ramping' activity observed empirically, suggesting that these two activity patterns emerge from the same mechanism.Keywords: perceptual decision-making; neural integrator; computational model of cortex; inhibitory neuron types; depressing synapses; ramping and stepping activity peer-reviewed)

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Mathematical Equivalence of Two Common Forms of Firing Rate Models of Neural Networks

Cited by 48 publications

References 13 publications

Optimality and Some of Its Discontents: Successes and Shortcomings of Existing Models for Binary Decisions

Optimality and Some of Its Discontents: Successes and Shortcomings of Existing Models for Binary Decisions

Properties of networks with partially structured and partially random connectivity

Cortical circuit-based lossless neural integrator for perceptual decision-making

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