Exponential

Contents

Exponential#

class braintools.conn.Exponential(distance_profile, weight=None, delay=None, **kwargs)#

Exponential distance-dependent connectivity for spatially organized neural networks.

This specialized connectivity class creates connections where the probability follows an exponential decay as a function of distance between neurons. This pattern is characteristic of many biological neural systems where connection probability drops off smoothly with distance, such as in hippocampal networks and certain cortical projections.

The connection probability follows:

\[P(d) = \exp\left(-\frac{d}{\lambda}\right) \quad \text{for } d \leq d_{\text{max}}\]

where:

  • \(d\) is the distance between neurons

  • \(\lambda\) is the spatial scale (decay length constant)

  • \(d_{\text{max}}\) is the maximum connection distance

Parameters:

  • distance_profile (ExponentialProfile) –

    An exponential distance profile that defines the spatial connection probability decay. Must be an instance of braintools.init.ExponentialProfile with:

    • scale: Characteristic decay length (λ) - larger values create longer-range connections

    • max_distance: Maximum distance for connections (optional)

    • normalize: Whether to normalize to unit maximum (optional)

  • weight (Initialization | float | int | ndarray | Array | Quantity | None) –

    Initializer for synaptic weights. Supports:

    • Constant values (e.g., 2.0 * u.nS)

    • Stochastic initializers (e.g., Normal, Uniform, LogNormal)

    • Distance-dependent weight distributions

    • None for unweighted connections

  • delay (Initialization | float | int | ndarray | Array | Quantity | None) –

    Initializer for synaptic transmission delays. Supports:

    • Constant values (e.g., 1.5 * u.ms)

    • Stochastic delay distributions

    • Distance-proportional delays

    • None for connections without explicit delays

  • **kwargs –

    Additional arguments passed to DistanceDependent, such as:

    • seed: Random seed for reproducible connection patterns

Returns:

Connection result containing pre/post indices, weights, delays, and metadata about the exponential connectivity pattern.

Return type:

ConnectionResult

Raises:

  • TypeError – If distance_profile is not an instance of ExponentialProfile.

  • ValueError – If neuron positions are not provided when generating connections.

Notes

  • The scale parameter (λ) determines how quickly connection probability decays:

    • At distance λ, probability is reduced to ~37% (1/e) of the maximum

    • Smaller scale values create more localized connectivity

    • Larger scale values allow longer-range connections

  • Exponential decay is generally slower than Gaussian decay at intermediate distances but faster at very short distances, making it suitable for modeling projections with moderate spatial extent

  • Setting max_distance improves computational efficiency by limiting the maximum connection range

  • Unlike Gaussian profiles, exponential profiles have no inflection point and decay monotonically

  • Connection generation is stochastic and requires a fixed seed for reproducibility

See also

Gaussian

Gaussian distance-dependent connectivity

DistanceDependent

General distance-dependent connectivity base class

braintools.init.ExponentialProfile

Exponential distance profile implementation

Examples

Basic exponential connectivity:

>>> import brainunit as u
>>> import numpy as np
>>> from braintools.conn import Exponential
>>> from braintools.init import ExponentialProfile, Constant
>>>
>>> # Create random 2D positions
>>> positions = np.random.uniform(0, 1000, (300, 2)) * u.um
>>>
>>> # Exponential connectivity with scale=150um
>>> conn = Exponential(
...     distance_profile=ExponentialProfile(
...         scale=150 * u.um,
...         max_distance=500 * u.um
...     ),
...     weight=Constant(1.5 * u.nS),
...     delay=Constant(1.0 * u.ms)
... )
>>>
>>> result = conn(
...     pre_size=300,
...     post_size=300,
...     pre_positions=positions,
...     post_positions=positions
... )

Short-range exponential connectivity:

>>> # Rapid decay for highly local connections
>>> local_conn = Exponential(
...     distance_profile=ExponentialProfile(
...         scale=75 * u.um,  # Short decay length
...         max_distance=225 * u.um
...     ),
...     weight=Constant(2.5 * u.nS)
... )

Long-range connections with variable weights:

>>> from braintools.init import Normal
>>>
>>> # Longer-range connectivity with stochastic weights
>>> long_range_conn = Exponential(
...     distance_profile=ExponentialProfile(
...         scale=300 * u.um,  # Long decay length
...         max_distance=900 * u.um
...     ),
...     weight=Normal(mean=1.0*u.nS, std=0.2*u.nS),
...     seed=42
... )

Distance-dependent delays:

>>> from braintools.init import DistanceModulated as DDDelay, LogNormal
>>>
>>> # Model conduction delays proportional to distance
>>> conn_with_delays = Exponential(
...     distance_profile=ExponentialProfile(
...         scale=200 * u.um,
...         max_distance=600 * u.um
...     ),
...     weight=Constant(1.0 * u.nS),
...     delay=DDDelay(lambda d: (0.5 * u.ms) + d / (300 * u.um/u.ms))
... )

Modeling hippocampal connectivity:

>>> # Exponential connectivity pattern typical of hippocampal networks
>>> positions_1d = np.linspace(0, 2000, 200).reshape(-1, 1) * u.um
>>>
>>> hippo_conn = Exponential(
...     distance_profile=ExponentialProfile(
...         scale=250 * u.um,  # Moderate spatial extent
...         max_distance=1000 * u.um
...     ),
...     weight=LogNormal(mean=1.5*u.nS, sigma=0.4),
...     delay=Normal(mean=2.0*u.ms, std=0.3*u.ms)
... )
>>>
>>> result = hippo_conn(
...     pre_size=200,
...     post_size=200,
...     pre_positions=positions_1d,
...     post_positions=positions_1d
... )
Contents