Computational Model: Visual to Default Network Pathways

Author

Danilo A Pereira, Ph.D.

Published

November 16, 2024

1 Introduction

This model formalizes the integration of semantic and spatial cognition using hierarchical Bayesian inference and the Drift-Diffusion Model (DDM). It aims to capture how visual stimuli are processed, how semantic and spatial information are encoded and integrated, and how decisions are made based on this information.

2 Model Components

2.1 Sensory Input Layer

2.1.1 Mathematical Definition

Visual stimuli S are presented as either objects O or scenes C.

S = \begin{cases} O_i, & \text{if object stimulus} \\ C_j, & \text{if scene stimulus} \end{cases}

2.1.2 Components

  • O_i: The i-th object stimulus
  • C_j: The j-th scene stimulus

2.2 Semantic Encoding

For an object O_i, infer semantic category \theta_S:

2.2.1 Bayesian Inference Equation

p(\theta_S|O_i) = \frac{p(O_i|\theta_S)p(\theta_S)}{\sum_{\theta_S} p(O_i|\theta_S)p(\theta_S)}

where:

  • p(O_i|\theta_S): Likelihood of object O_i given category \theta_S
  • p(\theta_S): Prior probability of category \theta_S

2.2.2 Example

2.2.2.1 Working Example

Assume three semantic categories: Instruments (\text{Inst}), Animals (\text{Anim}), Tools (\text{Tool}). An object O_i is a guitar.

Likelihoods:

p(O_i|\theta_S) = \begin{cases} 0.9, & \theta_S = \text{Inst} \\ 0.05, & \theta_S = \text{Anim} \\ 0.05, & \theta_S = \text{Tool} \end{cases}

Priors (assume uniform):

p(\theta_S) = \frac{1}{3}

Compute posteriors:

p(\text{Inst}|O_i) = \frac{0.9 \times \frac{1}{3}}{0.9 \times \frac{1}{3} + 0.05 \times \frac{1}{3} + 0.05 \times \frac{1}{3}} = 0.9

2.3 Spatial Encoding

For a scene C_j, infer spatial context \theta_C:

2.3.1 Bayesian Inference Equation

p(\theta_C|C_j) = \frac{p(C_j|\theta_C)p(\theta_C)}{\sum_{\theta_C} p(C_j|\theta_C)p(\theta_C)}

2.3.2 Example

2.3.2.1 Working Example

Two buildings: Building 1 (B1) and Building 2 (B2). Scene C_j is a living room from B1.

Likelihoods:

p(C_j|\theta_C) = \begin{cases} 0.8, & \theta_C = B1 \\ 0.2, & \theta_C = B2 \end{cases}

Priors (assume uniform):

p(\theta_C) = 0.5

Compute posteriors:

p(B1|C_j) = \frac{0.8 \times 0.5}{0.8 \times 0.5 + 0.2 \times 0.5} = 0.8

2.4 Integration of Semantic and Spatial Information

For cases where both O_i and C_j are relevant:

2.4.1 Joint Posterior

p(\theta_S,\theta_C|O_i,C_j) = \frac{p(O_i|\theta_S)p(C_j|\theta_C)p(\theta_S)p(\theta_C)}{\sum_{\theta_S,\theta_C} p(O_i|\theta_S)p(C_j|\theta_C)p(\theta_S)p(\theta_C)}

2.4.2 Example

Using previous likelihoods and priors. Compute joint posterior for \theta_S = \text{Inst} and \theta_C = B1:

Numerator:

N = p(O_i|\text{Inst})p(C_j|B1)p(\text{Inst})p(B1) = 0.9 \times 0.8 \times \frac{1}{3} \times 0.5 = 0.12

2.5 Decision-Making (Drift-Diffusion Model)

2.5.1 Evidence Accumulation

dx_t = vdt + \sigma dw_t

where:

  • x_t: Accumulated evidence at time t
  • v: Drift rate
  • \sigma: Noise parameter
  • dw_t: Wiener process increment

2.5.2 Drift Rate Calculations

For semantic decisions:

v = k(p(\theta_{\text{target}}|O_i) - p(\theta_{\text{distractor}}|O_i))

2.5.3 Decision Metrics

Accuracy:

P_{\text{correct}} = \frac{1}{1 + e^{-2v\alpha/\sigma^2}}

Response Time:

E[T] = \frac{\alpha}{v}\tanh(\frac{v\alpha}{\sigma^2})

2.5.4 Hierarchical Bayesian Parameter Estimation

Parameters like k, \sigma, \alpha can vary across individuals:

\begin{align*} k_i &\sim \text{Normal}(\mu_k, \sigma_k^2) \\ \sigma_i &\sim \text{Normal}(\mu_\sigma, \sigma_\sigma^2) \end{align*}

3 Implementation Steps

3.1 Step 1

Define Stimuli and Priors:

  • Specify objects O_i, scenes C_j
  • Define categories \theta_S, contexts \theta_C
  • Assign likelihoods and priors

3.2 Step 2

Compute Posteriors:

  • Calculate p(\theta_S|O_i)
  • Calculate p(\theta_C|C_j)
  • Integrate when needed

3.3 Step 3

Implement DDM:

  • Set up evidence accumulation
  • Calculate drift rates
  • Simulate decisions

4 Testing and Validation

4.1 Behavioral Predictions

4.1.1 Reaction Times

  • Predict response latencies
  • Compare with empirical data

4.1.2 Accuracy

  • Calculate error rates
  • Validate against human performance

4.2 Neural Correlates

4.2.1 Key Brain Regions

  • Semantic processing: lateral ventral occipital cortex
  • Spatial processing: medial visual cortex
  • Integration: right angular gyrus

5 Extensions

5.1 Learning Mechanisms

  • Update priors based on feedback
  • Incorporate temporal dynamics

5.2 Dynamic Neural Modeling

  • Stochastic differential equations
  • Neural population models

6 Ideas and Reasons Behind the Model

The choice of the Hierarchical Drift-Diffusion Model is motivated by several factors derived from the article:

6.1 Alignment with Decision-Making Processes

The study involves tasks where participants make semantic and spatial context decisions based on visual stimuli, requiring processing and decision-making under uncertainty (Methods, Study 1: Design and Procedure, pp.18-20). The Drift-Diffusion Model (DDM) is well-suited for modeling such two-choice decision tasks, capturing both accuracy and reaction times (Ratcliff & McKoon, 2008).

6.2 Behavioral Differences Between Tasks

The article reports that participants exhibited poorer accuracy and slower reaction times in the spatial context task compared to the semantic task (Behavioral Results, pp.3-4). This suggests differences in cognitive processing efficiency, which can be modeled by varying the drift rates between tasks in the HDDM.

6.3 Parallel Processing Streams

The existence of distinct neural pathways for semantic and spatial processing supports the idea that evidence accumulation rates may differ between tasks (Results, pp.5-14). Modeling these pathways as distinct drift rates reflects these neural differences.

6.4 Individual Differences and Hierarchical Structure

Variability among participants in neural and behavioral responses is acknowledged (Results, pp.10-14). A hierarchical model accounts for both group-level and individual-level parameters, capturing individual differences in cognitive processing.

7 Key Components and Mathematical Formulation

7.1 Evidence Accumulation Process

The DDM models decision-making as a stochastic process where evidence accumulates over time until a threshold is reached (Ratcliff & McKoon, 2008). The accumulation is described by:

dx(t) = vdt + \sigma dW(t)

  • v: Drift rate, representing the average speed of evidence accumulation.
  • \sigma: Noise parameter, representing within-trial variability.
  • dW(t): Increment of a Wiener process (i.e., Gaussian noise).

7.2 Decision Thresholds

Decisions occur when the accumulated evidence x(t) reaches one of two boundaries:

  • Upper boundary (a): Represents one choice (e.g., correct response).
  • Lower boundary (0): Represents the alternative choice (e.g., incorrect response).
  • Non-Decision Time (t_0): Accounts for time spent on processes not related to decision-making, such as sensory encoding and motor execution (Methods, MRI Data Acquisition, p.21).

7.3 Task-Specific Drift Rates

7.3.1 Semantic Task

Drift rate v_s: Expected to be higher due to the engagement of the lateral ventral occipital to frontotemporal DMN pathway supporting semantic judgments (Results, pp.5-7; Discussion, p.15).

7.3.2 Spatial Task

Drift rate v_c: May be lower, reflecting the involvement of the medial visual to medial temporal DMN pathway in spatial context judgments (Results, pp.5-7; Discussion, p.15).

7.4 Hierarchical Modeling of Parameters

Parameters are modeled hierarchically to capture variability across participants:

For participant i in task j:

v_{ij} = \mu_v + \alpha_i + \beta_j + \epsilon_{ij}

  • \mu_v: Overall mean drift rate.
  • \alpha_i: Random effect for participant i.
  • \beta_j: Fixed effect for task j (semantic or spatial).
  • \epsilon_{ij}: Residual error.

Similar hierarchical structures can be defined for boundary separation (a) and non-decision time (t_0) if necessary.

8 Relation to Article’s Concepts and Sources

8.1 Behavioral Outcomes Reflect Neural Pathways

The differences in reaction times and accuracy between tasks suggest that the efficiency of evidence accumulation (drift rate) is influenced by the distinct neural pathways engaged during semantic and spatial processing (Behavioral Results, pp.3-4; Results, pp.5-14).

8.2 Functional Differentiation in DMN

The article identifies functional subdivisions within the DMN that are associated with different cognitive processes (Results, pp.5-14; Discussion, pp.15-16). Modeling drift rates specific to each task reflects this functional differentiation.

8.3 Integration of Information

When semantic and spatial information can be integrated (e.g., in same-category buildings), participants’ performance may improve, suggesting interactions between pathways (Results, p.14; Discussion, pp.15-16). The model can incorporate interaction terms to capture these effects.