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📗 Explanation · understanding-oriented

Engineering vs. Biological Time

This explanation covers the two time‑unrolling conventions in DynVision, their mathematical equivalence, and how to convert between them.

Two Ways of Unrolling

Biological time — each feedforward connection has a propagation delay (\(\Delta_{FF} > 0\)), so activity propagates through the network depth over multiple timesteps. This matches cortical conduction velocities.

Engineering time — set \(\Delta_{FF} = 0\) so that activity propagates from input to output within a single timestep. The network becomes computationally more efficient but the temporal graph is mathematically equivalent when delays are converted correctly.

Engineering vs biological time unrolling

Figure: The same recurrent network unrolled in engineering time (left, \(\Delta_{FF}=0\)) and biological time (right, \(\Delta_{FF}=2\)). Signal flow through the network and time is identical in both conventions.

Delay Conversion Formulas

To switch from biological to engineering time in a network with skip and feedback connections:

\[\Delta_{SK}^{eng} = \Delta_{SK}^{bio} - dL \cdot \Delta_{FF}^{bio}$$ $$\Delta_{FB}^{eng} = \Delta_{FB}^{bio} + dL \cdot \Delta_{FF}^{bio}\]

where \(dL\) is the number of layers spanned by the connection.

Engineering‑time unrolling only works while \(\Delta_{SK}^{eng}\) is positive — i.e. skip connections must be as fast as, or faster than, the multi‑synaptic feedforward pathway.

Note: In practice these formulas may need a \(\pm 1\) offset depending on the order of operations. Skip delays typically need an increment because the source layer's computation for the current timestep has already been executed; feedback source computations for the same timestep have not.

Example

A DyRCNNx8 with full recurrence trained on CIFAR‑100 (30 timesteps, \(dt = 2\) ms):

Parameter Engineering Biological
\(\Delta_{FF}\) 0 ms 10 ms
\(\Delta_{RC}\) 6 ms 6 ms
\(\Delta_{SK}\) 2 ms 22 ms
\(\Delta_{FB}\) 30 ms 10 ms

Training in engineering time decreases epoch time by ~29 % and GPU memory from 2.39 GB to 2.13 GB.

Equivalence Validation

Equivalence of engineering and biological time

Figure: A DyRCNNx8 model trained in engineering time (with skip and feedback) and tested in both conventions produces identical temporal dynamics (shifted by \(\Delta_{FF}\)). This confirms that researchers can use the computationally more efficient engineering time for training while interpreting results in biological time.

The recurrence delay (\(\Delta_{RC} = 6\) ms) and time constant (\(\tau = 5\) ms) are independent of the unrolling convention; only the feedforward delay distinguishes them.

See Also