20,440 research outputs found
Geometrically stopped Markovian random growth processes and Pareto tails
Many empirical studies document power law behavior in size distributions of
economic interest such as cities, firms, income, and wealth. One mechanism for
generating such behavior combines independent and identically distributed
Gaussian additive shocks to log-size with a geometric age distribution. We
generalize this mechanism by allowing the shocks to be non-Gaussian (but
light-tailed) and dependent upon a Markov state variable. Our main results
provide sharp bounds on tail probabilities, a simple equation determining
Pareto exponents, and comparative statics. We present two applications: we show
that (i) the tails of the wealth distribution in a heterogeneous-agent dynamic
general equilibrium model with idiosyncratic investment risk are Paretian, and
(ii) a random growth model for the population dynamics of Japanese
municipalities is consistent with the observed Pareto exponent but only after
allowing for Markovian dynamics
Evidence accumulation in a Laplace domain decision space
Evidence accumulation models of simple decision-making have long assumed that
the brain estimates a scalar decision variable corresponding to the
log-likelihood ratio of the two alternatives. Typical neural implementations of
this algorithmic cognitive model assume that large numbers of neurons are each
noisy exemplars of the scalar decision variable. Here we propose a neural
implementation of the diffusion model in which many neurons construct and
maintain the Laplace transform of the distance to each of the decision bounds.
As in classic findings from brain regions including LIP, the firing rate of
neurons coding for the Laplace transform of net accumulated evidence grows to a
bound during random dot motion tasks. However, rather than noisy exemplars of a
single mean value, this approach makes the novel prediction that firing rates
grow to the bound exponentially, across neurons there should be a distribution
of different rates. A second set of neurons records an approximate inversion of
the Laplace transform, these neurons directly estimate net accumulated
evidence. In analogy to time cells and place cells observed in the hippocampus
and other brain regions, the neurons in this second set have receptive fields
along a "decision axis." This finding is consistent with recent findings from
rodent recordings. This theoretical approach places simple evidence
accumulation models in the same mathematical language as recent proposals for
representing time and space in cognitive models for memory.Comment: Revised for CB
Structure Learning in Coupled Dynamical Systems and Dynamic Causal Modelling
Identifying a coupled dynamical system out of many plausible candidates, each
of which could serve as the underlying generator of some observed measurements,
is a profoundly ill posed problem that commonly arises when modelling real
world phenomena. In this review, we detail a set of statistical procedures for
inferring the structure of nonlinear coupled dynamical systems (structure
learning), which has proved useful in neuroscience research. A key focus here
is the comparison of competing models of (ie, hypotheses about) network
architectures and implicit coupling functions in terms of their Bayesian model
evidence. These methods are collectively referred to as dynamical casual
modelling (DCM). We focus on a relatively new approach that is proving
remarkably useful; namely, Bayesian model reduction (BMR), which enables rapid
evaluation and comparison of models that differ in their network architecture.
We illustrate the usefulness of these techniques through modelling
neurovascular coupling (cellular pathways linking neuronal and vascular
systems), whose function is an active focus of research in neurobiology and the
imaging of coupled neuronal systems
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