"Reliable organisms from unreliable components" revisited: the linear drift, linear infinitesimal variance model of decision making

Diffusion models of decision making, in which successive samples of noisy evidence are accumulated to decision criteria, provide a theoretical solution to von Neumann's (1956) problem of how to increase the reliability of neural computation in the presence of noise. I introduce and evaluate a new neurally-inspired dual diffusion model, the linear drift, linear infinitesimal variance (LDLIV) model, which embodies three features often thought to characterize neural mechanisms of decision making. The accumulating evidence is intrinsically positively-valued, saturates at high intensities, and is accumulated for each alternative separately. I present explicit integral-equation predictions for the response time distribution and choice probabilities for the LDLIV model and compare its performance on two benchmark sets of data to three other models: the standard diffusion model and two dual diffusion model composed of racing Wiener processes, one between absorbing and reflecting boundaries and one with absorbing boundaries only. The LDLIV model and the standard diffusion model performed similarly to one another, although the standard diffusion model is more parsimonious, and both performed appreciably better than the other two dual diffusion models. I argue that accumulation of noisy evidence by a diffusion process and drift rate variability are both expressions of how the cognitive system solves von Neumann's problem, by aggregating noisy representations over time and over elements of a neural population. I also argue that models that do not solve von Neumann's problem do not address the main theoretical question that historically motivated research in this area.

A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes

Departing from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection–diffusion system for which very few solutions are known in exact form. The model considers also a deterministic source, and the coefficients of this equation are functions with sufficient regularity. A numerical scheme is designed to estimate the solutions of the initial-boundary-value problem. We prove rigorously that the numerical model is capable of preserving the main characteristics of the solutions of the stochastic model, that is, positivity, boundedness and monotonicity. The scheme has spatial symmetry, and it is theoretically analyzed for consistency, stability and convergence. Some numerical simulations are carried out in this work to assess the capability of the discrete model to preserve the main structural features of the solutions of the model. Moreover, a numerical study confirms the efficiency of the scheme, in agreement with the mathematical results obtained in this work.

First-passage-time distribution in a moving parabolic potential with spatial roughness

In this paper, we investigate the first-passage-time distribution (FPTD) within a time-dependent parabolic potential in the presence of roughness with two methods: the Kramers theory and a nonsingular integral equation. By spatially averaging, the rough potential is equivalent to the combination of an effective smooth potential and an effective diffusion coefficient. Based on the Kramers theory, we first obtain Kramers approximations (KAs) of FPTD for both smooth and rough potentials. As expected, KA is valid only for high barriers and small external forces, and generally applicable for high barriers in rough potentials. To overcome the shortcoming of KA, a probability asymptotic approximation (PAA) based on an integral equation is proposed, which uses the transient probability density function (PDF) of the natural boundary conditions instead of the absorbing boundary conditions. We find that PAA fits very well even for large external forces. This enables us to analytically solve the FPTD for large external forces and low barriers as a strong extension to KA. In addition, we show that in the presence of a rough potential, the PAA of FPTD is in good agreement with numerical simulations for low barrier potentials. The PAA makes it possible to investigate the first-passage problem with ultrafast varying potentials and short exiting time. Thus, KA and PAA are complementary in determining the FPTD both for various barriers and external forces. Finally, the mean first-passage time (MFPT) is studied, which illustrates that the PAA of MFPT is effective almost in the whole range of external forces, while the KA of MFPT is valid only for small external forces.

First-passage-time densities for time-non-homogeneous diffusion processes

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al. [4], [5].

A note on the Volterra integral equation for the first-passage-time probability density

In this paper we prove the validity of the Volterra integral equation for the evaluation of first-passage-time probability densities through varying boundaries, given by Buonocore et al. [1], for the case of diffusion processes not necessarily time-homogeneous. We study, specifically those processes that can be obtained from the Wiener process in the sense of [5]. A study of the kernel of the integral equation, in the same way as that by Buonocore et al. [1], is done. We obtain the boundaries for which closed-form solutions of the integral equation, without having to solve the equation, can be obtained. Finally, a few examples are given to indicate the actual use of our method.

A symmetry-based constructive approach to probability densities for one-dimensional diffusion processes

Special symmetry conditions on the transition p.d.f. of one-dimensional time-homogeneous diffusion processes with natural boundaries are investigated and exploited to derive closed-form results concerning the transition p.d.f.'s in the presence of absorbing and reflecting boundaries and the first-passage-time p.d.f. through time-dependent boundaries.

On the two-boundary first-crossing-time problem for diffusion processes

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.

On some first-crossing-time probabilities for a two-dimensional random walk with correlated components

For a two-dimensional random walk {X (n) = (X(n) 1, X(n) 2 )T, n ∈ ℕ0} with correlated components the first-crossing-time probability problem through unit-slope straight lines x 2 = x 1 - r(r = 0,1) is analysed. The p.g.f.'s for the first-crossing-time probabilities are expressed as solutions of a fourth-degree algebraic equation and are then exploited to obtain the first-crossing-time probabilities. Several additional results, including the mean first-crossing time and the probability of ultimate crossing, are also given.

Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes

Let X be a one-dimensional strong Markov process with continuous sample paths. Using Volterra-Stieltjes integral equation techniques we investigate Hölder continuity and differentiability of first passage time distributions of X with respect to continuous lower and upper moving boundaries. Under mild assumptions on the transition function of X we prove the existence of a continuous first passage time density to one-sided differentiable moving boundaries and derive a new integral equation for this density. We apply our results to Brownian motion and its nonrandom Markovian transforms, in particular to the Ornstein-Uhlenbeck process.

On the asymptotic behaviour of first-passage-time densities for one-dimensional diffusion processes and varying boundaries

Making use of the integral equations given in [1], [2] and [3], the asymptotic behaviour of the first-passage time (FPT) p.d.f.'s through certain time-varying boundaries, including periodic boundaries, is determined for a class of one-dimensional diffusion processes with steady-state density. Sufficient conditions are given for the cases both of single and of pairs of asymptotically constant and asymptotically periodic boundaries, under which the FPT densities asymptotically exhibit an exponential behaviour. Explicit expressions are then worked out for the processes that can be obtained from the Ornstein–Uhlenbeck process by spatial transformations. Some new asymptotic results for the FPT density of the Wiener process are finally proved, together with a few miscellaneous results.

On evaluations and asymptotic approximations of first-passage-time probabilities

The series expansion for the solution of the integral equation for the first-passage-time probability density function, obtained by resorting to the fixed point theorem, is used to achieve approximate evaluations for which error bounds are indicated. A different use of the fixed point theorem is then made to determine lower and upper bounds for asymptotic approximations, and to examine their range of validity.

A stochastic model of cancer growth subject to an intermittent treatment with combined effects: reduction in tumor size and rise in growth rate

A model of cancer growth based on the Gompertz stochastic process with jumps is proposed to analyze the effect of a therapeutic program that provides intermittent suppression of cancer cells. In this context, a jump represents an application of the therapy that shifts the cancer mass to a return state and it produces an increase in the growth rate of the cancer cells. For the resulting process, consisting in a combination of different Gompertz processes characterized by different growth parameters, the first passage time problem is considered. A strategy to select the inter-jump intervals is given so that the first passage time of the process through a constant boundary is as large as possible and the cancer size remains under this control threshold during the treatment. A computational analysis is performed for different choices of involved parameters. Finally, an estimation of parameters based on the maximum likelihood method is provided and some simulations are performed to illustrate the validity of the proposed procedure.

Theory of rapid force spectroscopy

In dynamic force spectroscopy, single (bio-)molecular bonds are actively broken to assess their range and strength. At low loading rates, the experimentally measured statistical distributions of rupture forces can be analysed using Kramers’ theory of spontaneous unbinding. The essentially deterministic unbinding events induced by the extreme forces employed to speed up full-scale molecular simulations have been interpreted in mechanical terms, instead. Here we start from a rigorous probabilistic model of bond dynamics to develop a unified systematic theory that provides exact closed-form expressions for the rupture force distributions and mean unbinding forces, for slow and fast loading protocols. Comparing them with Brownian dynamics simulations, we find them to work well also at intermediate pulling forces. This renders them an ideal companion to Bayesian methods of data analysis, yielding an accurate tool for analysing and comparing force spectroscopy data from a wide range of experiments and simulations.

Dynamic force spectroscopy is widely applied to probe molecular interactions by forcible bond breaking, but it currently lacks an analytical theory that spans the divide between experiment and simulation. Here, such a unified framework is developed and shown to be accurate for slow and fast loading.

A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model

A method to generate first passage times for a class of stochastic processes is proposed. It does not require construction of the trajectories as usually needed in simulation studies, but is based on an integral equation whose unknown quantity is the probability density function of the studied first passage times and on the application of the hazard rate method. The proposed procedure is particularly efficient in the case of the Ornstein-Uhlenbeck process, which is important for modeling spiking neuronal activity.

On some computational results for single neurons' activity modeling

The classical Ornstein-Uhlenbeck diffusion neuronal model is generalized by inclusion of a time-dependent input whose strength exponentially decreases in time. The behavior of the membrane potential is consequently seen to be modeled by a process whose mean and covariance classify, it as Gaussian-Markov. The effect of the input on the neuron's firing characteristics is investigated by comparing the firing probability densities and distributions for such a process with the corresponding ones of the Ornstein-Uhlenbeck model. All numerical results are obtained by implementation of a recently developed computational method.

Stochastic Dynamic Models of Response Time and Accuracy: A Foundational Primer

A large class of statistical decision models for performance in simple information processing tasks can be described by linear, first-order, stochastic differential equations (SDEs), whose solutions are diffusion processes. In such models, the first passage time for the diffusion process through a response criterion determines the time at which an observer makes a decision about the identity of a stimulus. Because the assumptions of many cognitive models lead to SDEs that are time inhomogeneous, classical methods for solving such first passage time problems are usually inapplicable. In contrast, recent integral equation methods often yield solutions to both the one-sided and the two-sided first passage time problems, even in the presence of time inhomogeneity. These methods, which are of particular relevance to the cognitive modeler, are described in detail, together with illustrative applications. Copyright 2000 Academic Press.

On the comparison of Feller and Ornstein-Uhlenbeck models for neural activity

Diffusion processes have been extensively used to describe membrane potential behavior. In this approach the interspike interval has a theoretical counterpart in the first-passage-time of the diffusion model employed. Since the mathematical complexity of the first-passage-time problem increases with attempts to make the models more realistic it seems useful to compare the features of different models in order to highlight their relative performance. In this paper we compare the Feller and Ornstein-Uhlenbeck models under three different criteria derived from the level of information available about their parameters. We conclude that the Feller model is preferable when complete knowledge of the characterizing parameters is assumed. On the other hand, when only limited information about the parameters is available, such as the mean firing time and the histogram shape, no advantage arises from using this more complex model.