Drift-diffusion Process Research Articles

Paradoxical Evidence Integration in Rapid Decision Processes

Decisions about noisy stimuli require evidence integration over time. Traditionally, evidence integration and decision making are described as a one-stage process: a decision is made when evidence for the presence of a stimulus crosses a threshold. Here, we show that one-stage models cannot explain psychophysical experiments on feature fusion, where two visual stimuli are presented in rapid succession. Paradoxically, the second stimulus biases decisions more strongly than the first one, contrary to predictions of one-stage models and intuition. We present a two-stage model where sensory information is integrated and buffered before it is fed into a drift diffusion process. The model is tested in a series of psychophysical experiments and explains both accuracy and reaction time distributions.

Visual fixations influence purchasing decisions less than choices between products

Event Abstract Back to Event Visual fixations influence purchasing decisions less than choices between products Ian Krajbich1, 2*, Dingchao Lu2, Colin Camerer2 and Antonio Rangel2 1 University of Zurich, Department of Economics, Switzerland 2 Caltech, Division of the Humanities and Social Sciences, United States We now know that visual fixations play a crucial role in two-item and multi-item choice by biasing the drift-diffusion choice process towards the item that is being fixated on. Here, we investigate whether visual fixations play a similar role in purchasing decisions. In the first stage of the experiment, subjects reveal their values for 50 different consumer goods using an incentive-compatible BDM auction. In the second stage (300 trials), subjects see one of the items and one of six random prices. Their decision is whether to purchase the item at that price. During these decisions we record subjects’ eye movements at 50Hz, as well as their choices and reaction times. We then use maximum likelihood estimation on the choices and reaction times to fit a modified drift-diffusion model where there is more evidence accumulated for an option when it is fixated than when it is not fixated. The naïve hypothesis was that purchasing decisions would obey the same drift-diffusion process as with two-item or multi-item choice, since the unchosen item(s) can be thought of as an opportunity cost, analogous to a real dollar cost. Instead, we find that the decision process in purchasing decisions is considerably different. Price fixations tend to be shorter and less variable than item fixations, indicating that subjects treat prices differently than items. Furthermore, while the structure of the best-fitting model is the same as before, the attention bias towards the fixated item is greatly diminished, resulting in better decisions. Keywords: Decision Theory, Eye Movements, reaction times Conference: Neural Coding, Decision-Making & Integration in Time, Rauischholzhausen, Germany, 26 Apr - 29 Apr, 2012. Presentation Type: Poster Presentation Topic: Neural Coding, Decision-Making & Integration in Time Citation: Krajbich I, Lu D, Camerer C and Rangel A (2012). Visual fixations influence purchasing decisions less than choices between products. Front. Neurosci. Conference Abstract: Neural Coding, Decision-Making & Integration in Time. doi: 10.3389/conf.fnins.2012.86.00006 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 12 Jan 2012; Published Online: 16 Jan 2012. * Correspondence: Dr. Ian Krajbich, University of Zurich, Department of Economics, Zurich, Switzerland, krajbich@gmail.com Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Ian Krajbich Dingchao Lu Colin Camerer Antonio Rangel Google Ian Krajbich Dingchao Lu Colin Camerer Antonio Rangel Google Scholar Ian Krajbich Dingchao Lu Colin Camerer Antonio Rangel PubMed Ian Krajbich Dingchao Lu Colin Camerer Antonio Rangel Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Unhedgeable shocks and statistical economic equilibrium

We develop a statistical concept of economic equilibrium as the stationary distribution of a random walk on the exchange equilibrium set (the contract set) of a pure exchange economy induced by unhedgeable shocks that perturb the economy from the exchange equilibrium set and subsequent disequilibrium trading that returns the economy to a new equilibrium. The Fokker–Planck equation for the resulting drift-diffusion process implies that the stationary distribution is independent of the size of the shock so that a small-disturbance limiting distribution is well defined. We present explicit solutions for the statistical equilibrium for the cases of quasilinear and Gorman-aggregatable Cobb–Douglas economies, and illustrate the results in the context of a generic dividend-discount model to emphasize the distinction between insurable risk and unhedgeable uncertainty in this context. The statistical equilibrium of income or wealth for quasilinear economies is described by an exponential Gibbs distribution. The statistical equilibrium income and wealth distributions for Gorman-aggregatable Cobb–Douglas economies can take a wider variety of forms, including power-law and gamma distributions. The statistical equilibria calculated for these examples suggest a close relation to widely observed statistical distributional regularities in real-world economies.

Analysis of mechanisms for platelet near-wall excess under arterial blood flow conditions

The concentration of platelets near the blood vessel wall is important because platelets survey the condition of the vessel wall and respond to injuries to it. Under arterial flow conditions, platelets are non-uniformly distributed across the vessel lumen and have a high concentration within a few microns of the vessel wall. This is believed to be a consequence of the complex motion of red blood cells which constitute a large fraction of the blood's volume. We use a novel lattice Boltzmann-immersed boundary method to simulate, in two dimensions, the motion of dense red blood cell suspensions and their effect on platelet-sized particles. We track the development of a red blood cell-free layer near the wall and the later development of the platelet near-wall excess. We find that the latter develops more quickly at high wall shear rates and that the magnitude of the excess and its proximity to the wall are dependent on haematocrit. Treating the simulation data as if it were generated by a drift–diffusion process, we find that the effective lateral platelet diffusivity depends strongly on lateral position; it has a magnitude of order of 10−6 cm2 s−1 over much of the lumen but drops to almost zero close to the wall. This large effective diffusivity over the core of the lumen combined with reduced space for platelets in this region because of the inward migration of red blood cells largely but not completely accounts for the observed platelet-concentration profiles. We present evidence for a highly localized red blood cell-induced platelet drift at the edge of the red cell-free layer and suggest a physical mechanism that may generate it.

Interval Timing by Long-Range Temporal Integration

Interval Timing by Long-Range Temporal Integration

Closed-Form Approximations of First-Passage Distributions for a Stochastic Decision-Making Model.

In free response choice tasks, decision making is often modeled as a first-passage problem for a stochastic differential equation. In particular, drift-diffusion processes with constant or time-varying drift rates and noise can reproduce behavioral data (accuracy and response-time distributions) and neuronal firing rates. However, no exact solutions are known for the first-passage problem with time-varying data. Recognizing the importance of simple closed-form expressions for modeling and inference, we show that an interrogation or cued-response protocol, appropriately interpreted, can yield approximate first-passage (response time) distributions for a specific class of time-varying processes used to model evidence accumulation. We test these against exact expressions for the constant drift case and compare them with data from a class of sigmoidal functions. We find that both the direct interrogation approximation and an error-minimizing interrogation approximation can capture a variety of distribution shapes and mode numbers but that the direct approximation, in particular, is systematically biased away from the correct free response distribution.

Two-stage dynamic signal detection: A theory of choice, decision time, and confidence.

The 3 most often-used performance measures in the cognitive and decision sciences are choice, response or decision time, and confidence. We develop a random walk/diffusion theory-2-stage dynamic signal detection (2DSD) theory-that accounts for all 3 measures using a common underlying process. The model uses a drift diffusion process to account for choice and decision time. To estimate confidence, we assume that evidence continues to accumulate after the choice. Judges then interrupt the process to categorize the accumulated evidence into a confidence rating. The model explains all known interrelationships between the 3 indices of performance. Furthermore, the model also accounts for the distributions of each variable in both a perceptual and general knowledge task. The dynamic nature of the model also reveals the moderating effects of time pressure on the accuracy of choice and confidence. Finally, the model specifies the optimal solution for giving the fastest choice and confidence rating for a given level of choice and confidence accuracy. Judges are found to act in a manner consistent with the optimal solution when making confidence judgments.

Dynamical Analysis of Bayesian Inference Models for the Eriksen Task

The Eriksen task is a classical paradigm that explores the effects of competing sensory inputs on response tendencies and the nature of selective attention in controlling these processes. In this task, conflicting flanker stimuli interfere with the processing of a central target, especially on short reaction time trials. This task has been modeled by neural networks and more recently by a normative Bayesian account. Here, we analyze the dynamics of the Bayesian models, which are nonlinear, coupled discrete time dynamical systems, by considering simplified, approximate systems that are linear and decoupled. Analytical solutions of these allow us to describe how posterior probabilities and psychometric functions depend on model parameters. We compare our results with numerical simulations of the original models and derive fits to experimental data, showing that agreements are rather good. We also investigate continuum limits of these simplified dynamical systems and demonstrate that Bayesian updating is closely related to a drift-diffusion process, whose implementation in neural network models has been extensively studied. This provides insight into how neural substrates can implement Bayesian computations.

The relative influences of priors and sensory evidence on an oculomotor decision variable during perceptual learning.

Choice behavior on simple sensory-motor tasks can exhibit trial-to-trial dependencies. For perceptual tasks, these dependencies reflect the influence of prior trials on choices that are also guided by sensory evidence, which is often independent across trials. Here we show that the relative influences of prior trials and sensory evidence on choice behavior can be shaped by training, such that prior influences are strongest when perceptual sensitivity to the relevant sensory evidence is weakest and then decline steadily as sensitivity improves. We trained monkeys to decide the direction of random-dot motion and indicate their decision with an eye movement. We characterized sequential dependencies by relating current choices to weighted averages of prior choices. We then modeled behavior as a drift-diffusion process, in which the weighted average of prior choices provided an additive offset to a decision variable that integrated incoming motion evidence to govern choice. The average magnitude of offset within individual training sessions declined steadily as the quality of the integrated motion evidence increased over many months of training. The trial-by-trial magnitude of offset was correlated with signals related to developing commands that generate the oculomotor response but not with neural activity in either the middle temporal area, which represents information about the motion stimulus, or the lateral intraparietal area, which represents the sensory-motor conversion. The results suggest that training can shape the relative contributions of expectations based on prior trends and incoming sensory evidence to select and prepare visually guided actions.

A neural network model of the Eriksen task: reduction, analysis, and data fitting.

We analyze a neural network model of the Eriksen task: a two-alternative forced-choice task in which subjects must correctly identify a central stimulus and disregard flankers that may or may not be compatible with it. We linearize and decouple the model, deriving a reduced drift-diffusion process with variable drift rate that describes the accumulation of net evidence in favor of either alternative, and we use this to analytically describe how accuracy and response time data depend on model parameters. Such analyses both assist parameter tuning in network models and suggest explanations of changing drift rates in terms of attention. We compare our results with numerical simulations of the full nonlinear model and with empirical data and show good fits to both with fewer parameters.

Lévy Processes, Saltatory Foraging, and Superdiffusion

It is well established that resource variability generated by spatial patchiness and turbulence is an important influence on the growth and recruitment of planktonic fish larvae. Empirical data show fractal-like prey distributions, and simulations indicate that scale-invariant foraging strategies may be optimal. Here we show how larval growth and recruitment in a turbulent environment can be formulated as a hitting time problem for a jump-diffusion process. We present two theoretical results. Firstly, if jumps are of a fixed size and occur as a Poisson process (embedded within a drift-diffusion), recruitment is effectively described by a diffusion process alone. Secondly, in the absence of diffusion, and for “patchy” jumps (of negative binomial size with Pareto inter-arrivals), the encounter process becomes superdiffusive. To synthesise these results we conduct a strategic simulation study where “patchy” jumps are embedded in a drift-diffusion process. We conclude that increasingly Levy-like predator foraging strategies can have a significantly positive effect on recruitment at the population level.

QUANTUM MONTE CARLO SIMULATIONS OF FERMIONS: A MATHEMATICAL ANALYSIS OF THE FIXED-NODE APPROXIMATION

The Diffusion Monte Carlo (DMC) method is a powerful strategy to estimate the ground state energy E0of an N-body Schrödinger Hamiltonian H = -½Δ + V with high accuracy. It consists of writing E0as the long-time limit of an expectation value of a drift-diffusion process with a source term, and numerically simulating this process by means of a collection of random walkers. As for a number of stochastic methods, a DMC calculation makes use of an importance sampling function ψIwhich hopefully approximates some ground state ψ0of H. In the fermionic case, it has been observed that the DMC method is biased, except in the special case when the nodal surfaces of ψIcoincide with those of a ground state of H. The approximation due to the fact that, in practice, the nodal surfaces of ψIdiffer from those of the ground states of H, is referred to as the Fixed Node Approximation (FNA). Our purpose in this paper is to provide a mathematical analysis of the FNA. We prove that, under convenient hypotheses, a DMC calculation performed with the importance sampling function ψI, provides an estimation of the infimum of the energy 〈ψ, Hψ〉 on the set of the fermionic test functions ψ that exactly vanish on the nodal surfaces of ψI.

The dynamics of choice among multiple alternatives

We consider neurally based models for decision-making in the presence of noisy incoming data. The two-alternative forced-choice task has been extensively studied, and in that case it is known that mutually inhibited leaky integrators in which leakage and inhibition balance can closely approximate a drift–diffusion process that is the continuum limit of the optimal sequential probability ratio test (SPRT). Here we study the performance of neural integrators in n ⩾ 2 alternative choice tasks and relate them to a multihypothesis sequential probability ratio test (MSPRT) that is asymptotically optimal in the limit of vanishing error rates. While a simple race model can implement this ‘max-vs-next’ MSPRT, it requires an additional computational layer, while absolute threshold crossing tests do not require such a layer. Race models with absolute thresholds perform relatively poorly, but we show that a balanced leaky accumulator model with an absolute crossing criterion can approximate a ‘max-vs-ave’ test that is intermediate in performance between the absolute and max-vs-next tests. We consider free and fixed time response protocols, and show that the resulting mean reaction times under the former and decision times for fixed accuracy under the latter obey versions of Hick's law in the low error rate range, and we interpret this in terms of information gained. Specifically, we derive relationships of the forms log ( n - 1 ) , log ( n ) , or log ( n + 1 ) depending on error rates, signal-to-noise ratio, and the test itself. We focus on linearized models, but also consider nonlinear effects of neural activities (firing rates) that are bounded below and show how they modify Hick's law.

Optimal Decisions: From Neural Spikes, through Stochastic Differential Equations, to Behavior

There is increasing evidence from in vivo recordings in monkeys trained to respond to stimuli by making left- or rightward eye movements, that firing rates in certain groups of neurons in oculo-motor areas mimic drift-diffusion processes, rising to a (fixed) threshold prior to movement initiation. This supplements earlier observations of psychologists, that human reaction-time and error-rate data can be fitted by random walk and diffusion models, and has renewed interest in optimal decision-making ideas from information theory and statistical decision theory as a clue to neural mechanisms. We review results from decision theory and stochastic ordinary differential equations, and show how they may be extended and applied to derive explicit parameter dependencies in optimal performance that may be tested on human and animal subjects. We then briefly describe a biophysically-based model of a pool of neurons in locus coeruleus, a brainstem nucleus implicated in widespread norepinephrine release. This neurotransmitter can effect transient gain changes in cortical circuits of the type that the abstract drift-diffusion analysis requires. We also describe how optimal gain schedules can be computed in the presence of time-varying noisy signals. We argue that a rational account of how neural spikes give rise to simple behaviors is beginning to emerge.

Study of direct current characteristics of carbon-doped GaInP/GaAs heterojunction bipolar transistor grown by solid source molecular beam epitaxy

This article reports the dc characteristics of a GaInP/GaAs heterojunction bipolar transistor (HBT) with carbon-doped GaAs base layer grown by solid source molecular beam epitaxy using carbon tetrabromide (CBr4) as p-type dopant precursor. Hydrofluoric acid (HF) was used to passivate the GaInP/GaAs HBTs. At base bias voltages below 0.9 V in the Gummel plot, the base current of large area devices after HF passivation, was greatly reduced indicating that the extrinsic base surface recombination current was significantly reduced. After HF passivation, detailed dc characterization of the device performance within the temperature range of 300–380 K was carried out, and the carrier transport properties were investigated. The base current and collector current ideality factors at 300 K were 1.12 and 1.01, respectively. This indicates that a space charge region recombination current is insignificant in the base. From the temperature dependent Gummel plot, the activation energies of the collector current and base current were obtained. For the collector current, the activation energy is 1.4 eV, which is close to the band gap of the GaAs base. This indicates that the collector current is determined by the drift-diffusion process. For the base current, the activation energy is also 1.4 eV, indicating that band-to-band recombination plays a dominant role in the base current. No trap-related recombination was observed in the base and collector currents, which further indicates the good base material quality in the HBT structure. The current gain versus collector current characteristics at different temperatures were also investigated and analyzed.

The Role of the Dielectric Barrier in Narrow Biological Channels: A Novel Composite Approach to Modeling Single-Channel Currents

A composite continuum theory for calculating ion current through a protein channel of known structure is proposed, which incorporates information about the channel dynamics. The approach is utilized to predict current through the Gramicidin A ion channel, a narrow pore in which the applicability of conventional continuum theories is questionable. The proposed approach utilizes a modified version of Poisson-Nernst-Planck (PNP) theory, termed Potential-of-Mean-Force-Poisson-Nernst-Planck theory (PMFPNP), to compute ion currents. As in standard PNP, ion permeation is modeled as a continuum drift-diffusion process in a self-consistent electrostatic potential. In PMFPNP, however, information about the dynamic relaxation of the protein and the surrounding medium is incorporated into the model of ion permeation by including the free energy of inserting a single ion into the channel, i.e., the potential of mean force along the permeation pathway. In this way the dynamic flexibility of the channel environment is approximately accounted for. The PMF profile of the ion along the Gramicidin A channel is obtained by combining an equilibrium molecular dynamics (MD) simulation that samples dynamic protein configurations when an ion resides at a particular location in the channel with a continuum electrostatics calculation of the free energy. The diffusion coefficient of a potassium ion within the channel is also calculated using the MD trajectory. Therefore, except for a reasonable choice of dielectric constants, no direct fitting parameters enter into this model. The results of our study reveal that the channel response to the permeating ion produces significant electrostatic stabilization of the ion inside the channel. The dielectric self-energy of the ion remains essentially unchanged in the course of the MD simulation, indicating that no substantial changes in the protein geometry occur as the ion passes through it. Also, the model accounts for the experimentally observed saturation of ion current with increase of the electrolyte concentration, in contrast to the predictions of standard PNP theory.

Elevated Temperature Characteristics of Carbon-Doped GaInP/GaAs Heterojunction Bipolar Transistor Grown by Solid Source Molecular Beam Epitaxy

ABSTRACTThis paper reports the characteristics of GaInP/GaAs heterojunction bipolar transistor (HBT) with carbon-doped GaAs base layer grown by solid source molecular beam epitaxy (SSMBE) using carbon tetrabromide (CBr4) as p-type dopant precursor. Hydrofluoric acid (HF) was used to passivate the GaInP/GaAs HBTs. At base bias voltages below 0.8V in the Gummel plot, the base current of large-area devices after HF treatment was greatly reduced. This indicates that the extrinsic base surface recombination current was greatly reduced. After HF treatment, detailed DC characterization of the device performance from 300K to 380K was carried out and the carrier transport properties were investigated. The base current and collector current ideality factors at 300K were 1.12 and 1.01, respectively. This indicates that the space- charge region recombination current in the base is insignificant. From the temperature- dependent Gummel plot, the activation energies of collector current and base current were obtained. For the collector current, the activation energy is 1.4eV, which is close to the bandgap of the GaAs base. This indicates that the collector current is determined by the drift-diffusion process, in which an energy barrier of the same magnitude as the base bandgap is to be overcome by electrons before they reach the collector. For the base current, the activation energy is also 1.4eV, which is close to the bandgap of GaAs, indicating that band-to-band recombination plays a dominant role in the base current. No trap-related recombination was observed for the base and collector currents, which further indicates the high quality carbon-doped GaAs base material for the HBT structures.

To the uniqueness problem for nonlinear elliptic equations

We prove existence and uniqueness of solutions u∈W 1,2(Ω)∩L ∞(Ω) to equations of the form − ∑ i=1 n ∂ ∂x i ρ(u)b i x, ∂u ∂x +a x,u, ∂u ∂x =0, x∈Ω. Our nonstandard assumptions on the coefficients are such that log ρ(u) is concave and a( x, u, ξ)/ ρ( u) is increasing in u. Such assumptions are natural in view of drift diffusion processes for example in semiconductors and chemotaxis.

Drift diffusion and Einstein relation for electrons in silicon subjected to a high electric field

We present a theory that makes an explicit connection between scattering-limited Ohmic mobility and quantum-emission-limited saturation velocity. The theory is applied to electrons in bulk silicon by taking a quantum equal to the energy of an optical phonon. A modification in the mean-free path because of this quantum emission is indicated, an effect that appears only in the high-field regime. This modification is shown to lead to electric-field-induced degradation of the diffusion coefficient. The theory presented agrees well with the drift-diffusion experimental data and empirical relations used in modeling devices. The theory makes connections with an alternate description in terms of electron temperature under ac and dc conditions. Because drift-diffusion processes are central in performance evaluation of submicron-scale devices in which high fields are necessarily present, these results contribute significantly to reshaping thinking processes in the high-field regime.

A test of the Fokker–Planck approach for the description of semiconductor transport properties

We apply the Fokker–Planck approach (FPA), where electronic transport is modeled as a drift-diffusion process in energy space, in the calculation of transport properties of bulk Si for a moderately high electric field along the 〈111〉 direction. The main goal is to test the FPA by comparing the experimental and Monte Carlo data with the FPA results for a well-known semiconductor. As a consequence, a reliable basis for the discussion of the validity and limitations of the FPA would be provided in a realistic model of a well-known semiconductor.