RC5 Interval Algebra

Overview

fSTG Toolkit uses the RC5 (Region Connection Calculus, 5 relations) to encode how the connectivity of a brain area changes between two consecutive time steps. Each temporal edge in a spatio-temporal graph carries an RC5 label that concisely describes the topological relationship between two connectivity patterns.

RC5 is a subset of Allen’s interval algebra, restricted to five mutually exclusive and jointly exhaustive relations between two regions (connectivity patterns).

The Five Relations

Let A and B be the sets of brain areas connected to a given region at time steps t and t+1 respectively.

Relation	Symbol	Meaning	Connectivity change
Equal	EQ	A = B	The connectivity pattern is unchanged
Proper Part	PP	A ⊂ B	Connectivity gained new areas (growth)
Proper Part (inverse)	PPi	B ⊂ A	Connectivity lost areas (shrinkage)
Partial Overlap	PO	A ∩ B ≠ ∅, A ≠ B	Some areas retained, some gained, some lost
Disconnected	DC	A ∩ B = ∅	Complete reorganisation; no overlap

Interpretation for Brain Connectivity

These relations provide a semantically meaningful vocabulary for describing connectivity dynamics:

• EQ — The brain area maintains exactly the same functional connections across the two time points. This indicates a stable connectivity state.

• PP — The area’s connectivity pattern grows: all previously connected areas remain, and additional areas become connected. This may signal network expansion or recruitment.

• PPi — The connectivity pattern shrinks: some connections are lost while none are gained. This may reflect network pruning or disengagement.

• PO — The pattern partially reorganises: some connections are retained while others are replaced. This is the most common relation in dynamic data.

• DC — The connectivity pattern undergoes complete reorganisation with no overlap to the previous time step. This usually indicates unrelated regions.

Usage in Code

RC5 transitions are represented by the fstg_toolkit.graph.RC5 enum:

from fstg_toolkit.graph import RC5

# Accessing a relation
print(RC5.EQ)    # RC5.EQ
print(RC5.PPi)   # RC5.PPi

# Lookup by name
rc5 = RC5.from_name("PO")  # → RC5.PO

# Iterate over all relations
for relation in RC5:
    print(relation)

Temporal edges in a fstg_toolkit.graph.SpatioTemporalGraph carry the RC5 label in their rc5 attribute:

for u, v, data in graph.edges(data=True):
    if data.get('type') == 'temporal':
        print(f"{u} → {v}: {data['rc5']}")

References

• Randell, et al. (1992) A spatial logic based on regions and connection., Proceedings of the 3rd International Conference on Knowledge Representation and Reasoning, pp. 165–176.