Fast Solutions for the First-Passage Distribution of Diffusion Models with Space-Time-Dependent Drift Functions and Time-Dependent Boundaries

Boehm, Udo; Cox, Sonja; Gantner, Gregor; Stevenson, Rob

doi:10.31234/osf.io/maurt

Cited by 3 publications

(6 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, while many interesting models are straightforward to simulate, often researchers want to go the other way: from observed data to infer the most likely parameters. For all but the simplest models, such likelihood functions are analytically intractable, and hence previous approaches required computationally costly simulations and/or lacked flexibility in applying such methods to different scenarios [4,45,52,53].…”

Section: /33mentioning

confidence: 99%

“…In particular, while many interesting models are straightforward to simulate, often researchers want to go the other way: from the observed data to infer the most likely parameters. For all but the simplest models, such likelihood functions are analytically intractable, and hence previous approaches required computationally costly simulations and/or lacked flexibility in applying such methods to different scenarios (Boehm et al, 2021; Palestro et al, 2019; Shinn et al, 2020; Turner & Sederberg, 2014; Turner & Van Zandt, 2018). We recently developed a novel approach using artificial neural networks which can, given sufficient training data, approximate likelihoods for a large class of SSM variants, thereby amortizing the cost and enabling rapid, efficient and flexible inference (Fengler et al, 2021).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Beyond Drift Diffusion Models: Fitting a broad class of decision and RL models with HDDM

Fengler

Frank

Bera

et al. 2022

Preprint

View full text Add to dashboard Cite

Computational modeling has become an central aspect of research in the cognitive neurosciences. As the field matures, it is increasingly important to move beyond standard models to quantitatively assess models with richer dynamics that may better reflect underlying cognitive and neural processes. For example, sequential sampling models (SSMs) are a general class of models of decision making intended to capture processes jointly giving rise to reaction time distributions and choice data in n-alternative paradigms. A number of model variations are of theoretical interest, but empirical data analysis has historically been tied to a small subset for which likelihood functions are analytically tractable. Advances in methods designed for likelihood-free inference have recently made it computationally feasible to consider a much larger spectrum of sequential sampling models. In addition, recent work has motivated the combination of SSMs with reinforcement learning (RL) models, which had historically been considered in separate literatures. Here we provide a significant addition to the widely used HDDM python toolbox and include a tutorial for how users can easily fit and assess a (user extensible) wide variety of SSMs, and how they can be combined with RL models. The extension comes batteries included, including model visualization tools, posterior predictive checks, and ability to link trial-wise neural signals with model parameters via hierarchical Bayesian regression.

show abstract

Section: /33mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Beyond Drift Diffusion Models: Fitting a broad class of decision and RL models with HDDM

Fengler

Frank

Bera

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

“…In a parallel publication [BCGS21], aimed at the neuroscientific community, we provide further numerical experiments and code. In particular, in [BCGS21] we apply the Crank-Nicolson method to approximate the solution e to (3.1). In the examples we consider it appears that the Crank-Nicolson method leads to similar convergence as the minimal residual method.…”

Section: Introductionmentioning

confidence: 99%

“…Although theoretical results for the minimal residual method are stronger, Crank-Nicolson may be preferred in practice as it is easier to implement. We refer to [BCGS21] for further details. For normed linear spaces E and F, by L(E, F) we denote the normed linear space of bounded linear mappings E → F, and by L iso (E, F) its subset of boundedly invertible linear mappings E → F.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Efficient numerical approximation of a non-regular Fokker--Planck equation associated with first-passage time distributions

Boehm,

Cox,

Gantner

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker-Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations.

show abstract

The Mean First-Passage Time and Stochastic Resonance of a Stochastic Tumor-immune Model via BPNN Method

Zhang

Yang

et al. 2023

Preprint

View full text Add to dashboard Cite

The Mean First-Passage Time (MFPT) and Stochastic Resonance (SR) of a stochastic tumor-immune model with noises perturbation are discussed in this paper. Firstly, considering environmental perturbation, Gaussian white noise and Gaussian colored noise are introduced into a tumor growth model under immune surveillance. As the follows, the long-time evolution of the tumor characterized by the Stationary Probability Density (SPD) and MFPT are obtained in theory on the basis of the Approximated Fokker-Planck Equation (AFPE). Herein the recurrence of the tumor from the extinction state to the tumor-present state is more concerned in this paper. A higher efficient algorithm of Back-Propagation Neural Network (BPNN) is applied in order to testify the correction of the theoretic SPD and MFPT. With the existence of weak signal, the functional relationship between Signal-to-Noise Ratio (SNR), noise intensities and correlation time is also studied. These results show that both multiplicative Gaussian colored noise and additive Gaussian white noise can promote the extinction of tumors, and the multiplicative Gaussian colored noise will lead to the resonance-like peak on MFPT curves, while the increasing intensity of additive Gaussian white noise will cause the minimum of MFPT. In addition, the correlation times are negatively correlated with MFPT. As for the SNR, we find the intensities of Gaussian white noise and Gaussian colored noise, as well as their correlation intensity will induce SR. Especially, SNR is monotonously increased in the case of Gaussian white noise with the change of the correlation time, however, SNR displays the point of inflection in other cases. At last, the optimal parameters in BPNN structure are analyzed for MFPT from three aspects: the penalty factors, the number of neural network layers and the number of nodes in each layer.

show abstract

Fast Solutions for the First-Passage Distribution of Diffusion Models with Space-Time-Dependent Drift Functions and Time-Dependent Boundaries

Cited by 3 publications

References 41 publications

Beyond Drift Diffusion Models: Fitting a broad class of decision and RL models with HDDM

Beyond Drift Diffusion Models: Fitting a broad class of decision and RL models with HDDM

Efficient numerical approximation of a non-regular Fokker--Planck equation associated with first-passage time distributions

The Mean First-Passage Time and Stochastic Resonance of a Stochastic Tumor-immune Model via BPNN Method

Contact Info

Product

Resources

About