Geometry, SFMs and FEP

[Item: 7b35a6f1-6434-4659-ba9d-43d6b101efad]

https://www.tangramvision.com/blog/the-deceptively-asymmetric-unit-sphere

Interesting tour of differential geometry. Deep reinforcement learning approaches similar manifold oriented problems as in SFMs and active inference (the latter has non Euclidean geometry because probability distribution parameter space is not Euclidean)

https://arxiv.org/pdf/1210.2354v2 describes the two univariate Gaussian case in detail

Friston’s FEP says action choice follows shortest path ie geodesics in those spaces. Maybe links to fluidity: low fluidity is choosing long geodesics between states, and high fluidity is choosing short geodesics. So fluidity is a signature of information geometry of rest state manifold, which is the internal model Q in FEP

☕️☕️

I asked Claude to elaborate some hypotheses, here's what I got:

Let me analyze these concepts more deeply and propose some testable hypotheses. I'll call the ResearchAgent to gather additional context about recent developments in these areas.

Let me break this down into testable hypotheses focusing on the relationship between information geometry, geodesics, and cognitive fluidity.

Key Hypotheses:

1. Geodesic Length and Cognitive Efficiency

• H1: Systems with shorter geodesic paths between cognitive states exhibit higher behavioral flexibility

• Test Method:

  • Measure response times in task-switching paradigms
  • Compare path lengths in probability distribution space
  • Correlate geodesic lengths with behavioral adaptability metrics

1. Information Geometric Signatures of Fluidity

• H2: High-fluidity states correspond to optimal information geometric structures

• Test Method:

  • Map state transitions in parameter space
  • Measure Fisher information metric at different fluidity levels
  • Compare curvature properties between high/low fluidity states

1. Rest State Manifold Properties

• H3: The geometry of rest state manifolds predicts transition capabilities

• Test Method:

  • Characterize manifold properties (curvature, topology)
  • Measure state transition efficiencies
  • Correlate manifold properties with behavioral flexibility

1. Active Inference and Path Optimization

• H4: Systems following FEP principles naturally converge to geodesic paths

• Test Method:

  • Compare actual paths taken vs. theoretical geodesics
  • Measure free energy along different possible paths
  • Analyze convergence patterns in learning tasks

Experimental Approaches:

• Computational modeling of manifold structures

• Behavioral experiments measuring state transitions

• Neural imaging to map state space trajectories

• Information geometric analysis of decision processes

This framework could help quantify and predict cognitive flexibility based on underlying geometric principles.

can we narrow this down for schizophrenia?

I'll help reformulate these hypotheses specifically for schizophrenia, focusing on how information geometry and geodesics might relate to positive and negative symptoms.

1. Geodesic Distortion in Positive Symptoms

• H1: Positive symptoms (hallucinations, delusions) represent abnormally short geodesics between unrelated cognitive states

• Test Method:

  • Compare geodesic distances between perceptual/belief states in:
  • Schizophrenia patients with active positive symptoms
  • Healthy controls
  • Patients in remission
  • Measure how easily unrelated concepts become connected (semantic priming tasks)
  • Quantify "jump distances" in thought processes using natural language processing

1. Extended Geodesics in Negative Symptoms

• H2: Negative symptoms (avolition, anhedonia) represent abnormally long geodesics between normally adjacent states

• Test Method:

  • Measure transition times between:
  • Intention and action
  • Stimulus and emotional response
  • Social cues and social responses
  • Compare path lengths in reward-motivation networks
  • Track effort-reward decision-making trajectories

1. Rest State Manifold Topology in Schizophrenia

• H3: The rest state manifold in schizophrenia shows distinct topological features

  • For positive symptoms: excessive local connections, creating "shortcuts"
  • For negative symptoms: reduced connectivity, creating "elongated paths"

• Test Method:

  • fMRI analysis of default mode network topology
  • Graph theoretical analysis of functional connectivity
  • Compare manifold curvature properties between symptom types

1. Active Inference and Precision

• H4: Symptom severity correlates with specific distortions in precision weighting of predictions

• Test Method:

  • Measure precision of predictions in sensory tasks
  • Compare belief updating patterns in probabilistic learning
  • Track changes in confidence levels during decision-making

1. Information Geometric Markers

• H5: Different symptom profiles show distinct signatures in information geometry

• Test Method:

  • Map Fisher information metrics for different symptom profiles
  • Compare Riemannian distance measures between cognitive states
  • Analyze probability distribution parameters during task performance

Specific Experimental Approaches:

1. Clinical Measurements

• Track symptom severity using PANSS or SANS/SAPS scales

• Correlate with geometric measures

• Longitudinal tracking of geometric changes with symptom changes

1. Cognitive Tasks

• Probabilistic learning tasks

• Sensory prediction tasks

• Social cognition paradigms

• Effort-based decision-making tasks

1. Neuroimaging

• High-density EEG to track state transitions

• fMRI for spatial mapping of state spaces

• Dynamic causal modeling to quantify connection strengths

1. Computational Modeling

• Build generative models of state transitions

• Simulate effects of different precision weightings

• Model impact of altered geodesic distances

These hypotheses could help:

• Identify geometric biomarkers for symptom profiles

• Develop early warning systems for symptom emergence

• Guide intervention strategies based on geometric principles

• Inform personalized treatment approaches

The key innovation here is viewing schizophrenia symptoms through an information geometric lens, potentially offering new ways to quantify and treat both positive and negative symptoms based on their distinct geometric signatures.