MADE is a Python package that implements neural attractor networks on various manifolds, providing tools for both Continuous Attractor Networks (CANs) and Quasiperiodic Attractor Networks (QANs).
MADE provides implementations of neural networks that can maintain continuous families of stable states on various manifolds. The package supports:
- Line (R¹) - One-dimensional linear space
- Ring (S¹) - One-dimensional periodic space
- Plane (R²) - Two-dimensional Euclidean space
- Torus (T²) - Two-dimensional doubly periodic space
- Cylinder (R × S¹) - Mixed periodic/non-periodic space
- Möbius Band - Non-orientable surface with one periodic dimension
- Sphere (S²) - Two-dimensional surface embedded in 3D
pip install ManifoldAttractorsThe python package is called made, so for example you'll import with:
from made import manifolds
from made.can import CANWe provide three notebooks to demonstrate the usage of the package.
In 1_visualize_connectivity.ipynb we show how one can visualize a Metric over a Manifold as well as
how to visualize a CAN's connectivity and it's state (as a bump on the manifold).
In 2_visualize_activity_manifold.ipynb we show how to create a large number of simulations, each with
the CAN's bump at a different point. Then we use Isomap to create a 3D visualization of the activity manifold
in state space.
In 3_visualize_qans_integration.ipynb we show how to create a QAN and visualize it's (offset) connectivity.
We then use the QAN to generate a trajectory over the manifold and visualize how the QAN reconstructs it.
A CAN is created by defining a Manifold and a Metric. Several manifolds and metrics have been defined in
made.manifolds and made.metrics:
| Manifold | Metric | Dimension | Periodic |
|---|---|---|---|
| Line | Euclidean | 1 | False |
| Ring | Periodic Euclidean | 1 | True |
| Plane | Euclidean | 2 | False, False |
| Torus | Periodic Euclidean | 2 | True, True |
| Cylinder | Periodic Euclidean | 2 | False, True |
| Möbius Band | Möbius | 1 | False, True |
| Sphere | Great Circle | 3* | True, True, True |
* Sphere is a 2D manifold, but we only consider it as the unit sphere embedded in 3D.
The same principles can be used to define new manifolds and metrics to construct CANs with diverse manifold topologies.
A QAN uses multiple copies of a CAN, each with different offsets applied during computation of the connectivity. This sets up a push-pull mechanism that allows the QAN to track movement on the manifold by modulating the activity of each CAN based on the position of the bump on the manifold and a velocity vector indicating how it should move.