Tutorial 3: Distance-Dependent Connectivity Patterns

Tutorial 3: Distance-Dependent Connectivity Patterns#

Colab

This tutorial explores distance-dependent connectivity patterns that are fundamental to spatial neural networks. In biological neural systems, connection probability and strength often depend on the physical distance between neurons.

Topics Covered#

  1. Distance profiles for spatial neural connectivity

  2. Basic profiles: Gaussian, Exponential, PowerLaw, Linear, Step

  3. Advanced profiles: Sigmoid, DoG, Logistic, Bimodal, MexicanHat

  4. Profile composition: arithmetic operations, clipping, transformations

  5. Pipe operator for functional composition

  6. Real-world neuroscience applications

Installation and Setup#

# Install braintools if needed
# !pip install braintools brainunit matplotlib numpy scipy

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.gridspec import GridSpec
import brainunit as u
from braintools import init

# Set random seed for reproducibility
np.random.seed(42)

# Configure matplotlib
plt.rcParams['figure.figsize'] = (12, 4)
plt.rcParams['font.size'] = 10

1. Introduction to Distance-Dependent Connectivity#

Biological Motivation#

In the brain, neurons are spatially organized, and their connectivity follows specific patterns:

  • Local connectivity: Neurons close together are more likely to connect

  • Distance decay: Connection probability decreases with distance

  • Lateral inhibition: Nearby neurons may inhibit slightly distant ones

  • Long-range connections: Some neurons connect across large distances

Distance Profiles#

A distance profile defines how connection properties vary with spatial distance:

probability = profile.probability(distances)
weight_scaling = profile.weight_scaling(distances)

Key Concepts#

  1. Connection probability: Likelihood of forming a connection

  2. Weight scaling: Strength modulation based on distance

  3. Spatial units: Using physical distances (μm, mm, etc.)

  4. Profile composition: Combining multiple patterns

# Helper function to visualize profiles
def visualize_profile(profile, max_distance=200, title="Distance Profile", unit=u.um):
    """
    Visualize a distance profile.
    """
    distances = np.linspace(0, max_distance, 500) * unit
    probabilities = profile.probability(distances)
    weights = profile.weight_scaling(distances)
    
    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    
    # Probability
    axes[0].plot(distances.mantissa, probabilities, linewidth=2)
    axes[0].fill_between(distances.mantissa, probabilities, alpha=0.3)
    axes[0].set_xlabel(f'Distance ({unit})')
    axes[0].set_ylabel('Connection Probability')
    axes[0].set_title(f'{title} - Probability')
    axes[0].grid(alpha=0.3)
    axes[0].set_ylim([0, 1.1])
    
    # Weight scaling
    axes[1].plot(distances.mantissa, weights, linewidth=2, color='orange')
    axes[1].fill_between(distances.mantissa, weights, alpha=0.3, color='orange')
    axes[1].set_xlabel(f'Distance ({unit})')
    axes[1].set_ylabel('Weight Scaling Factor')
    axes[1].set_title(f'{title} - Weight Scaling')
    axes[1].grid(alpha=0.3)
    
    plt.tight_layout()
    plt.show()
    
    return distances, probabilities, weights

2. Basic Distance Profiles#

2.1 Gaussian Profile#

The Gaussian profile creates bell-shaped connectivity with peak at distance 0:

\[p(d) = \exp\left(-\frac{d^2}{2\sigma^2}\right)\]

Parameters:

  • sigma: Standard deviation (controls spread)

  • max_distance: Optional cutoff distance

Use cases:

  • Local excitatory connections

  • Smooth distance decay

  • Cortical columns

# Create Gaussian profile
gaussian = init.GaussianProfile(
    sigma=50.0 * u.um,
    max_distance=200.0 * u.um
)

distances, probs, weights = visualize_profile(gaussian, title="Gaussian Profile")

print(f"Gaussian Profile Properties:")
print(f"  Probability at 0 μm: {probs[0]:.3f}")
print(f"  Probability at 50 μm (1σ): {probs[250]:.3f}")
print(f"  Half-width at half-max: ~{50 * np.sqrt(2 * np.log(2)):.1f} μm")
../_images/417567e9ec944911a53cee65ba861f2d69ecdd076feda4d73a88863a92e571a4.png 
Gaussian Profile Properties:
  Probability at 0 μm: 1.000
  Probability at 50 μm (1σ): 0.134
  Half-width at half-max: ~58.9 μm

Effect of Sigma Parameter#

# Compare different sigma values
sigmas = [20.0, 50.0, 100.0]
colors = ['red', 'blue', 'green']

fig, ax = plt.subplots(1, 1, figsize=(10, 5))

distances = np.linspace(0, 200, 500) * u.um

for sigma, color in zip(sigmas, colors):
    profile = init.GaussianProfile(sigma=sigma * u.um)
    probs = profile.probability(distances)
    ax.plot(distances.mantissa, probs, linewidth=2, label=f'σ = {sigma} μm', color=color)
    ax.fill_between(distances.mantissa, probs, alpha=0.2, color=color)

ax.set_xlabel('Distance (μm)', fontsize=12)
ax.set_ylabel('Connection Probability', fontsize=12)
ax.set_title('Gaussian Profile: Effect of Sigma', fontsize=14)
ax.legend(fontsize=11)
ax.grid(alpha=0.3)

plt.tight_layout()
plt.show()

print("💡 Larger sigma → broader connectivity")
../_images/594285a1271e165c5e2a94adce6d343f85394be0b47ce35ade993aeda4bab08f.png 
💡 Larger sigma → broader connectivity

2.2 Exponential Profile#

The Exponential profile creates monotonic decay:

\[p(d) = \exp\left(-\frac{d}{\lambda}\right)\]

Parameters:

  • decay_constant (λ): Distance constant for decay

  • max_distance: Optional cutoff

Use cases:

  • Long-range connections

  • Axonal arbor patterns

  • Distance-dependent decay

# Create Exponential profile
exponential = init.ExponentialProfile(
    decay_constant=75.0 * u.um,
    max_distance=300.0 * u.um
)

visualize_profile(exponential, max_distance=300, title="Exponential Profile")

print(f"\nExponential Profile Properties:")
print(f"  Probability at 0: 1.000")
print(f"  Probability at λ (75 μm): {np.exp(-1):.3f} ≈ 0.368")
print(f"  Probability at 2λ (150 μm): {np.exp(-2):.3f} ≈ 0.135")
print(f"  Half-distance: {75 * np.log(2):.1f} μm")
../_images/575567ab041ee10fbb1fdcd1bce3c9035032c818b6fe663185e6237fff197abd.png 
Exponential Profile Properties:
  Probability at 0: 1.000
  Probability at λ (75 μm): 0.368 ≈ 0.368
  Probability at 2λ (150 μm): 0.135 ≈ 0.135
  Half-distance: 52.0 μm

Gaussian vs Exponential#

# Compare Gaussian and Exponential
fig, axes = plt.subplots(1, 2, figsize=(14, 5))

distances = np.linspace(0, 200, 500) * u.um

gaussian_prof = init.GaussianProfile(sigma=50.0 * u.um)
exponential_prof = init.ExponentialProfile(decay_constant=50.0 * u.um)

gauss_prob = gaussian_prof.probability(distances)
exp_prob = exponential_prof.probability(distances)

# Linear scale
axes[0].plot(distances.mantissa, gauss_prob, linewidth=2, label='Gaussian (σ=50)', color='blue')
axes[0].plot(distances.mantissa, exp_prob, linewidth=2, label='Exponential (λ=50)', color='red')
axes[0].set_xlabel('Distance (μm)', fontsize=11)
axes[0].set_ylabel('Probability', fontsize=11)
axes[0].set_title('Linear Scale', fontsize=12)
axes[0].legend()
axes[0].grid(alpha=0.3)

# Log scale
axes[1].semilogy(distances.mantissa, gauss_prob, linewidth=2, label='Gaussian (σ=50)', color='blue')
axes[1].semilogy(distances.mantissa, exp_prob, linewidth=2, label='Exponential (λ=50)', color='red')
axes[1].set_xlabel('Distance (μm)', fontsize=11)
axes[1].set_ylabel('Probability (log scale)', fontsize=11)
axes[1].set_title('Log Scale', fontsize=12)
axes[1].legend()
axes[1].grid(alpha=0.3)

plt.suptitle('Gaussian vs Exponential Decay', fontsize=14, y=1.02)
plt.tight_layout()
plt.show()

print("\n📊 Key differences:")
print("  Gaussian: Symmetric, bell-shaped, faster tail decay")
print("  Exponential: Monotonic, linear in log-space, heavier tails")
../_images/51d4b5c3b4aed3ea7533302c79b4ef6182ab0961a1f55aabf4c832954e54b286.png 
📊 Key differences:
  Gaussian: Symmetric, bell-shaped, faster tail decay
  Exponential: Monotonic, linear in log-space, heavier tails

2.3 Power Law Profile#

The Power Law profile follows:

\[p(d) = d^{-\alpha}\]

Parameters:

  • exponent (α): Power law exponent

  • min_distance: Avoid division by zero

  • max_distance: Optional cutoff

Use cases:

  • Scale-free networks

  • Long-range connections

  • Brain-wide connectivity

# Create Power Law profile
powerlaw = init.PowerLawProfile(
    exponent=2.0,
    min_distance=10.0 * u.um,
    max_distance=500.0 * u.um
)

visualize_profile(powerlaw, max_distance=500, title="Power Law Profile")

# Compare different exponents
fig, ax = plt.subplots(1, 1, figsize=(10, 5))

distances = np.linspace(10, 500, 500) * u.um
exponents = [0.5, 1.0, 2.0, 3.0]

for exp in exponents:
    profile = init.PowerLawProfile(exponent=exp, min_distance=10.0 * u.um)
    probs = profile.probability(distances)
    ax.loglog(distances.mantissa, probs, linewidth=2, label=f'α = {exp}')

ax.set_xlabel('Distance (μm, log scale)', fontsize=11)
ax.set_ylabel('Probability (log scale)', fontsize=11)
ax.set_title('Power Law Profile: Different Exponents', fontsize=12)
ax.legend()
ax.grid(alpha=0.3, which='both')

plt.tight_layout()
plt.show()

print("\n💡 Power law properties:")
print("  - Linear in log-log space")
print("  - Larger α → steeper decay")
print("  - Heavy tails (scale-free)")
../_images/cf0ea17f35fe0707ad79e15cbecd18cf485317ce3a8cffaf515f8b5f27babdf3.png ../_images/2c1f7d9850500d6b1e02808da2ce135be03be43376386239e4899407c70bb782.png 
💡 Power law properties:
  - Linear in log-log space
  - Larger α → steeper decay
  - Heavy tails (scale-free)

2.4 Linear Profile#

The Linear profile decreases linearly:

\[p(d) = \max\left(0, 1 - \frac{d}{d_{max}}\right)\]

Parameters:

  • max_distance: Distance at which probability reaches 0

Use cases:

  • Simple distance decay

  • Hard cutoffs

  • Piecewise connectivity

# Create Linear profile
linear = init.LinearProfile(max_distance=150.0 * u.um)

visualize_profile(linear, max_distance=200, title="Linear Profile")

print("\nLinear Profile Properties:")
print("  - Simplest decay pattern")
print("  - Hard cutoff at max_distance")
print("  - Uniform rate of decay")
../_images/34e7bca98ce790d3b29efb8f1fa9c95516e5aeedd751e4d1d4814a721a374447.png 
Linear Profile Properties:
  - Simplest decay pattern
  - Hard cutoff at max_distance
  - Uniform rate of decay

2.5 Step Profile#

The Step profile creates binary connectivity regions:

\[\begin{split}p(d) = \begin{cases} p_{in} & d \leq d_{threshold} \\ p_{out} & d > d_{threshold} \end{cases}\end{split}\]

Parameters:

  • threshold: Distance threshold

  • inside_prob: Probability inside threshold

  • outside_prob: Probability outside threshold

Use cases:

  • Local vs long-range separation

  • Two-population connectivity

  • Binary spatial domains

# Create Step profile
step = init.StepProfile(
    threshold=100.0 * u.um,
    inside_prob=0.9,
    outside_prob=0.1
)

visualize_profile(step, max_distance=200, title="Step Profile")

# Compare different step configurations
fig, ax = plt.subplots(1, 1, figsize=(10, 5))

distances = np.linspace(0, 200, 1000) * u.um

configs = [
    (100.0, 1.0, 0.0, 'Hard cutoff'),
    (100.0, 0.8, 0.2, 'Soft transition'),
    (100.0, 1.0, 0.5, 'Long-range maintained'),
]

for threshold, p_in, p_out, label in configs:
    profile = init.StepProfile(threshold * u.um, p_in, p_out)
    probs = profile.probability(distances)
    ax.plot(distances.mantissa, probs, linewidth=2, label=label)

ax.set_xlabel('Distance (μm)', fontsize=11)
ax.set_ylabel('Connection Probability', fontsize=11)
ax.set_title('Step Profile Configurations', fontsize=12)
ax.legend()
ax.grid(alpha=0.3)

plt.tight_layout()
plt.show()
../_images/b76b24a21365adc89701c8ee2ed5857ab044f544bd53312ebc2d38e6334b4ef4.png ../_images/1606fc6d590f0034534820f4f729ea8199629e9f8028a3c6e96980fe57885b56.png

3. Advanced Distance Profiles#

3.1 Sigmoid Profile#

The Sigmoid profile creates smooth S-shaped transitions:

\[p(d) = \frac{1}{1 + e^{k(d - d_0)}}\]

Parameters:

  • midpoint: Distance at which probability is 0.5

  • slope: Steepness of transition

  • max_distance: Optional cutoff

Use cases:

  • Smooth transitions

  • Threshold-like behavior

  • Gradual connectivity changes

# Create Sigmoid profile
sigmoid = init.SigmoidProfile(
    midpoint=100.0 * u.um,
    slope=0.05,
    max_distance=300.0 * u.um
)

visualize_profile(sigmoid, max_distance=300, title="Sigmoid Profile")

# Compare different slopes
fig, ax = plt.subplots(1, 1, figsize=(10, 5))

distances = np.linspace(0, 250, 500) * u.um
slopes = [0.01, 0.05, 0.1, 0.2]

for slope in slopes:
    profile = init.SigmoidProfile(midpoint=100.0 * u.um, slope=slope)
    probs = profile.probability(distances)
    ax.plot(distances.mantissa, probs, linewidth=2, label=f'slope = {slope}')

ax.axvline(100, color='red', linestyle='--', alpha=0.5, label='Midpoint')
ax.axhline(0.5, color='red', linestyle='--', alpha=0.5)
ax.set_xlabel('Distance (μm)', fontsize=11)
ax.set_ylabel('Connection Probability', fontsize=11)
ax.set_title('Sigmoid Profile: Effect of Slope', fontsize=12)
ax.legend()
ax.grid(alpha=0.3)

plt.tight_layout()
plt.show()

print("\n💡 Larger slope → sharper transition (approaches step function)")
../_images/dcb345013ccf364c5194337758995ee0d03e33c51e808396f1d7c0a8711da57d.png ../_images/b374d8a9311823a00584451a6fe6b3be33806e7fa818d5aa64d8ec234897b688.png 
💡 Larger slope → sharper transition (approaches step function)

3.2 Difference of Gaussians (DoG) Profile#

The DoG profile creates center-surround patterns:

\[p(d) = A_c e^{-\frac{d^2}{2\sigma_c^2}} - A_s e^{-\frac{d^2}{2\sigma_s^2}}\]

Parameters:

  • sigma_center: Center Gaussian width

  • sigma_surround: Surround Gaussian width (> sigma_center)

  • amplitude_center: Center amplitude

  • amplitude_surround: Surround amplitude

Use cases:

  • Lateral inhibition

  • Center-surround receptive fields

  • Competitive dynamics

# Create DoG profile
dog = init.DoGProfile(
    sigma_center=30.0 * u.um,
    sigma_surround=90.0 * u.um,
    amplitude_center=1.0,
    amplitude_surround=0.5,
    max_distance=250.0 * u.um
)

visualize_profile(dog, max_distance=250, title="Difference of Gaussians Profile")

# Visualize components
fig, ax = plt.subplots(1, 1, figsize=(10, 5))

distances = np.linspace(0, 250, 500) * u.um

# Components
center_gaussian = init.GaussianProfile(sigma=30.0 * u.um)
surround_gaussian = init.GaussianProfile(sigma=90.0 * u.um)

center = center_gaussian.probability(distances)
surround = 0.5 * surround_gaussian.probability(distances)
dog_result = dog.probability(distances)

ax.plot(distances.mantissa, center, 'b--', linewidth=2, label='Center Gaussian', alpha=0.7)
ax.plot(distances.mantissa, surround, 'r--', linewidth=2, label='Surround Gaussian', alpha=0.7)
ax.plot(distances.mantissa, dog_result, 'g-', linewidth=3, label='DoG (Difference)')
ax.axhline(0, color='black', linestyle='-', linewidth=0.5)
ax.fill_between(distances.mantissa, 0, dog_result, alpha=0.3, color='green')

ax.set_xlabel('Distance (μm)', fontsize=11)
ax.set_ylabel('Response', fontsize=11)
ax.set_title('DoG Components and Result', fontsize=12)
ax.legend()
ax.grid(alpha=0.3)

plt.tight_layout()
plt.show()

print("\n🧠 DoG creates lateral inhibition:")
print("  - Strong excitation at center")
print("  - Inhibition at intermediate distances")
print("  - Common in visual cortex receptive fields")
../_images/c845a372f9b9a3f6733020009a4ece080d5e653522ce739c21d4a0ed6ad9a98e.png ../_images/8fcc75dca3eef42b48039bebf716056a9a82aee0a11d08c53b5e97a286cf3b2a.png 
🧠 DoG creates lateral inhibition:
  - Strong excitation at center
  - Inhibition at intermediate distances
  - Common in visual cortex receptive fields

3.3 Mexican Hat Profile#

The Mexican Hat profile is the second derivative of a Gaussian:

\[p(d) = A \cdot \left(1 - \frac{d^2}{\sigma^2}\right) e^{-\frac{d^2}{2\sigma^2}}\]

Parameters:

  • sigma: Width parameter

  • amplitude: Scaling factor

  • max_distance: Optional cutoff

Use cases:

  • Center-surround patterns

  • Lateral inhibition

  • Self-organizing maps

# Create Mexican Hat profile
mexican_hat = init.MexicanHatProfile(
    sigma=50.0 * u.um,
    amplitude=1.0,
    max_distance=300.0 * u.um
)

visualize_profile(mexican_hat, max_distance=300, title="Mexican Hat Profile")

# Compare with DoG
fig, ax = plt.subplots(1, 1, figsize=(10, 5))

distances = np.linspace(0, 300, 500) * u.um

mh_profile = init.MexicanHatProfile(sigma=50.0 * u.um)
dog_profile = init.DoGProfile(sigma_center=30.0 * u.um, sigma_surround=90.0 * u.um)

mh_prob = mh_profile.probability(distances)
dog_prob = dog_profile.probability(distances)

ax.plot(distances.mantissa, mh_prob, linewidth=2, label='Mexican Hat')
ax.plot(distances.mantissa, dog_prob, linewidth=2, label='DoG')
ax.axhline(0, color='black', linestyle='-', linewidth=0.5)

ax.set_xlabel('Distance (μm)', fontsize=11)
ax.set_ylabel('Response', fontsize=11)
ax.set_title('Mexican Hat vs DoG', fontsize=12)
ax.legend()
ax.grid(alpha=0.3)

plt.tight_layout()
plt.show()

print("\n📊 Mexican Hat properties:")
print("  - Single parameter (sigma) controls shape")
print("  - More regular than DoG")
print("  - Ricker wavelet in signal processing")
../_images/81a8981101c3ad3e4a7d257562329407a5e2d1b04603dccb03ac01f3a4cb9ce8.png ../_images/0bfaf4500240cc8746c1114752a3002252872166d0cffe4b25beb014e7a24c4b.png 
📊 Mexican Hat properties:
  - Single parameter (sigma) controls shape
  - More regular than DoG
  - Ricker wavelet in signal processing

3.4 Bimodal Profile#

The Bimodal profile has two peaks:

\[p(d) = A_1 e^{-\frac{(d-c_1)^2}{2\sigma_1^2}} + A_2 e^{-\frac{(d-c_2)^2}{2\sigma_2^2}}\]

Parameters:

  • sigma1, sigma2: Widths of two peaks

  • center1, center2: Locations of peaks

  • amplitude1, amplitude2: Heights of peaks

Use cases:

  • Local + long-range connections

  • Multi-scale connectivity

  • Hierarchical networks

# Create Bimodal profile
bimodal = init.BimodalProfile(
    sigma1=30.0 * u.um,
    sigma2=50.0 * u.um,
    center1=0.0 * u.um,
    center2=200.0 * u.um,
    amplitude1=1.0,
    amplitude2=0.6,
    max_distance=400.0 * u.um
)

visualize_profile(bimodal, max_distance=400, title="Bimodal Profile")

# Show components
fig, ax = plt.subplots(1, 1, figsize=(10, 5))

distances = np.linspace(0, 400, 500) * u.um

# Individual Gaussians
gauss1 = init.GaussianProfile(sigma=30.0 * u.um)
gauss2 = init.GaussianProfile(sigma=50.0 * u.um)

prob1 = gauss1.probability(distances)
prob2 = 0.6 * gauss2.probability(distances - 200.0 * u.um + distances[0])
bimodal_prob = bimodal.probability(distances)

ax.plot(distances.mantissa, prob1, '--', linewidth=2, label='Local peak', alpha=0.7)
ax.plot(distances.mantissa, prob2, '--', linewidth=2, label='Long-range peak', alpha=0.7)
ax.plot(distances.mantissa, bimodal_prob, linewidth=3, label='Bimodal (sum)')

ax.set_xlabel('Distance (μm)', fontsize=11)
ax.set_ylabel('Connection Probability', fontsize=11)
ax.set_title('Bimodal Profile Components', fontsize=12)
ax.legend()
ax.grid(alpha=0.3)

plt.tight_layout()
plt.show()

print("\n🔗 Bimodal connectivity:")
print("  - Local connections (first peak)")
print("  - Long-range connections (second peak)")
print("  - Common in cortical circuits")
../_images/f1361729c2ddf99f42bf7b089ef0ec6f765e9202f87332081182af9fc3e791c8.png ../_images/0536bb8668eab5e90d03b735183095bae244ac2d680435f9860829433a12349e.png 
🔗 Bimodal connectivity:
  - Local connections (first peak)
  - Long-range connections (second peak)
  - Common in cortical circuits

3.5 Logistic Profile#

The Logistic profile is similar to sigmoid:

\[p(d) = \frac{1}{1 + e^{k(d - d_0)}}\]

Parameters:

  • growth_rate: Controls decay speed

  • midpoint: Half-probability distance

Use cases:

  • Growth-like patterns

  • Smooth transitions

  • Similar to sigmoid

# Create Logistic profile
logistic = init.LogisticProfile(
    growth_rate=0.05,
    midpoint=100.0 * u.um,
    max_distance=300.0 * u.um
)

visualize_profile(logistic, max_distance=300, title="Logistic Profile")
../_images/66b3fe24412453637fd2982534e7ab688e24f28fd58d3ebff5ab090e1480b25e.png 
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        3.75403722e-03, 3.64327465e-03, 3.53576852e-03, 3.43142377e-03,
        3.33014807e-03, 3.23185174e-03, 3.13644771e-03, 3.04385139e-03,
        2.95398067e-03, 2.86675578e-03, 2.78209927e-03, 2.69993593e-03,
        2.62019274e-03, 2.54279877e-03, 2.46768518e-03, 2.39478509e-03,
        2.32403359e-03, 2.25536765e-03, 2.18872606e-03, 2.12404940e-03,
        2.06127999e-03, 2.00036180e-03, 1.94124046e-03, 1.88386318e-03,
        1.82817868e-03, 1.77413723e-03, 1.72169049e-03, 1.67079159e-03,
        1.62139498e-03, 1.57345647e-03, 1.52693315e-03, 1.48178337e-03,
        1.43796670e-03, 1.39544389e-03, 1.35417684e-03, 1.31412856e-03,
        1.27526315e-03, 1.23754576e-03, 1.20094257e-03, 1.16542073e-03,
        1.13094838e-03, 1.09749458e-03, 1.06502930e-03, 1.03352338e-03,
        1.00294855e-03, 9.73277334e-04, 9.44483080e-04, 9.16539919e-04,
        8.89422738e-04, 8.63107166e-04, 8.37569546e-04, 8.12786920e-04,
        7.88737001e-04, 7.65398160e-04, 7.42749406e-04, 7.20770364e-04,
        6.99441258e-04, 6.78742897e-04, 6.58656652e-04, 6.39164446e-04,
        6.20248732e-04, 6.01892480e-04, 5.84079164e-04, 5.66792743e-04,
        5.50017650e-04, 5.33738776e-04, 5.17941459e-04, 5.02611467e-04,
        4.87734989e-04, 4.73298622e-04, 4.59289358e-04, 4.45694572e-04,
        4.32502013e-04, 4.19699789e-04, 4.07276362e-04, 3.95220531e-04,
        3.83521430e-04, 3.72168511e-04, 3.61151536e-04, 3.50460573e-04,
        3.40085981e-04, 3.30018403e-04, 3.20248762e-04, 3.10768244e-04,
        3.01568298e-04, 2.92640628e-04, 2.83977178e-04, 2.75570134e-04,
        2.67411910e-04, 2.59495148e-04, 2.51812703e-04, 2.44357643e-04,
        2.37123243e-04, 2.30102974e-04, 2.23290500e-04, 2.16679674e-04,
        2.10264530e-04, 2.04039277e-04, 1.97998297e-04, 1.92136138e-04,
        1.86447509e-04, 1.80927273e-04, 1.75570449e-04, 1.70372201e-04,
        1.65327836e-04, 1.60432800e-04, 1.55682674e-04, 1.51073169e-04,
        1.46600124e-04, 1.42259500e-04, 1.38047379e-04, 1.33959956e-04,
        1.29993541e-04, 1.26144553e-04, 1.22409516e-04, 1.18785057e-04,
        1.15267903e-04, 1.11854879e-04, 1.08542901e-04, 1.05328979e-04,
        1.02210211e-04, 9.91837803e-05, 9.62469527e-05, 9.33970764e-05,
        9.06315773e-05, 8.79479578e-05, 8.53437941e-05, 8.28167340e-05,
        8.03644951e-05, 7.79848623e-05, 7.56756863e-05, 7.34348813e-05,
        7.12604232e-05, 6.91503479e-05, 6.71027493e-05, 6.51157778e-05,
        6.31876386e-05, 6.13165900e-05, 5.95009416e-05, 5.77390534e-05,
        5.60293336e-05, 5.43702380e-05, 5.27602675e-05, 5.11979678e-05,
        4.96819275e-05, 4.82107771e-05, 4.67831874e-05, 4.53978687e-05]))

4. Profile Composition#

One of the most powerful features of distance profiles is the ability to compose them using arithmetic operations and transformations.

4.1 Arithmetic Operations#

Profiles support standard arithmetic operations:

  • Addition (+): Combine connectivity patterns

  • Subtraction (-): Create inhibitory patterns

  • Multiplication (*): Scale or modulate

  • Division (/): Normalize or ratio

# Create base profiles
local = init.GaussianProfile(sigma=30.0 * u.um)
distant = init.GaussianProfile(sigma=100.0 * u.um)

# Compose profiles
combined_add = local + distant * 0.3
combined_sub = local - distant * 0.2
scaled = local * 0.5

# Visualize
fig, axes = plt.subplots(2, 2, figsize=(14, 10))
axes = axes.flatten()

distances = np.linspace(0, 250, 500) * u.um

# Addition
prob_local = local.probability(distances)
prob_distant = distant.probability(distances)
prob_add = combined_add.probability(distances)

axes[0].plot(distances.mantissa, prob_local, '--', label='Local', alpha=0.7)
axes[0].plot(distances.mantissa, prob_distant * 0.3, '--', label='Distant × 0.3', alpha=0.7)
axes[0].plot(distances.mantissa, prob_add, linewidth=2, label='Sum')
axes[0].set_title('Addition: Local + Distant')
axes[0].set_xlabel('Distance (μm)')
axes[0].set_ylabel('Probability')
axes[0].legend()
axes[0].grid(alpha=0.3)

# Subtraction
prob_sub = combined_sub.probability(distances)
axes[1].plot(distances.mantissa, prob_local, '--', label='Local', alpha=0.7)
axes[1].plot(distances.mantissa, prob_distant * 0.2, '--', label='Distant × 0.2', alpha=0.7)
axes[1].plot(distances.mantissa, prob_sub, linewidth=2, label='Difference')
axes[1].axhline(0, color='black', linestyle='-', linewidth=0.5)
axes[1].set_title('Subtraction: Local - Distant')
axes[1].set_xlabel('Distance (μm)')
axes[1].set_ylabel('Probability')
axes[1].legend()
axes[1].grid(alpha=0.3)

# Scaling
prob_scaled = scaled.probability(distances)
axes[2].plot(distances.mantissa, prob_local, '--', label='Original', alpha=0.7)
axes[2].plot(distances.mantissa, prob_scaled, linewidth=2, label='Scaled × 0.5')
axes[2].set_title('Scaling: Profile × 0.5')
axes[2].set_xlabel('Distance (μm)')
axes[2].set_ylabel('Probability')
axes[2].legend()
axes[2].grid(alpha=0.3)

# Complex composition
complex_profile = (local * 0.8) + (distant * 0.3) - (init.GaussianProfile(sigma=60.0 * u.um) * 0.1)
prob_complex = complex_profile.probability(distances)
axes[3].plot(distances.mantissa, prob_complex, linewidth=2, color='purple')
axes[3].fill_between(distances.mantissa, prob_complex, alpha=0.3, color='purple')
axes[3].set_title('Complex: 0.8×Local + 0.3×Distant - 0.1×Medium')
axes[3].set_xlabel('Distance (μm)')
axes[3].set_ylabel('Probability')
axes[3].grid(alpha=0.3)

plt.suptitle('Profile Composition with Arithmetic Operations', fontsize=14, y=1.00)
plt.tight_layout()
plt.show()

print("\n✨ Arithmetic operations enable complex connectivity patterns!")
../_images/528eafae4af860b74dff0844ddcd8113ed646c58c8497d3d33e3f26423767e97.png 
✨ Arithmetic operations enable complex connectivity patterns!

4.2 Clipping Transformations#

The .clip() method bounds profile values:

clipped = profile.clip(min_val=0.1, max_val=0.9)

# Create profile and apply clipping
original = init.GaussianProfile(sigma=60.0 * u.um)
clipped_min = original.clip(min_val=0.3, max_val=None)
clipped_max = original.clip(min_val=None, max_val=0.7)
clipped_both = original.clip(min_val=0.2, max_val=0.8)

# Visualize
fig, axes = plt.subplots(2, 2, figsize=(14, 10))
axes = axes.flatten()

distances = np.linspace(0, 250, 500) * u.um

# Original
prob_original = original.probability(distances)
axes[0].plot(distances.mantissa, prob_original, linewidth=2)
axes[0].fill_between(distances.mantissa, prob_original, alpha=0.3)
axes[0].set_title('Original Profile')
axes[0].set_xlabel('Distance (μm)')
axes[0].set_ylabel('Probability')
axes[0].grid(alpha=0.3)
axes[0].set_ylim([0, 1])

# Clipped min
prob_min = clipped_min.probability(distances)
axes[1].plot(distances.mantissa, prob_min, linewidth=2)
axes[1].fill_between(distances.mantissa, prob_min, alpha=0.3)
axes[1].axhline(0.3, color='red', linestyle='--', label='Min = 0.3')
axes[1].set_title('Clipped (min=0.3)')
axes[1].set_xlabel('Distance (μm)')
axes[1].set_ylabel('Probability')
axes[1].legend()
axes[1].grid(alpha=0.3)
axes[1].set_ylim([0, 1])

# Clipped max
prob_max = clipped_max.probability(distances)
axes[2].plot(distances.mantissa, prob_max, linewidth=2)
axes[2].fill_between(distances.mantissa, prob_max, alpha=0.3)
axes[2].axhline(0.7, color='red', linestyle='--', label='Max = 0.7')
axes[2].set_title('Clipped (max=0.7)')
axes[2].set_xlabel('Distance (μm)')
axes[2].set_ylabel('Probability')
axes[2].legend()
axes[2].grid(alpha=0.3)
axes[2].set_ylim([0, 1])

# Clipped both
prob_both = clipped_both.probability(distances)
axes[3].plot(distances.mantissa, prob_both, linewidth=2)
axes[3].fill_between(distances.mantissa, prob_both, alpha=0.3)
axes[3].axhline(0.2, color='red', linestyle='--', alpha=0.7, label='Min = 0.2')
axes[3].axhline(0.8, color='red', linestyle='--', alpha=0.7, label='Max = 0.8')
axes[3].set_title('Clipped (min=0.2, max=0.8)')
axes[3].set_xlabel('Distance (μm)')
axes[3].set_ylabel('Probability')
axes[3].legend()
axes[3].grid(alpha=0.3)
axes[3].set_ylim([0, 1])

plt.suptitle('Profile Clipping', fontsize=14, y=1.00)
plt.tight_layout()
plt.show()

print("\n✂️ Clipping use cases:")
print("  - Ensure minimum connectivity")
print("  - Cap maximum connection strength")
print("  - Create plateaus in connectivity")
../_images/ada6d316c8ef212895458e43aaf8f993984be8960587165cd0cc0419d109dfd6.png 
✂️ Clipping use cases:
  - Ensure minimum connectivity
  - Cap maximum connection strength
  - Create plateaus in connectivity

4.3 Apply Transformations#

The .apply() method applies arbitrary functions:

transformed = profile.apply(lambda x: x ** 2)

# Create profile and apply transformations
base = init.GaussianProfile(sigma=60.0 * u.um)

# Different transformations
squared = base.apply(lambda x: x ** 2)
sqrt = base.apply(lambda x: np.sqrt(x))
inverted = base.apply(lambda x: 1 - x)
sigmoid_transform = base.apply(lambda x: 1 / (1 + np.exp(-10 * (x - 0.5))))

# Visualize
fig, axes = plt.subplots(2, 2, figsize=(14, 10))
axes = axes.flatten()

distances = np.linspace(0, 200, 500) * u.um

profiles = [
    (squared, 'Squared: x²', 'Power law'),
    (sqrt, 'Square root: √x', 'Flattened'),
    (inverted, 'Inverted: 1-x', 'Anti-correlation'),
    (sigmoid_transform, 'Sigmoid: 1/(1+e^(-10(x-0.5)))', 'Binarized'),
]

for ax, (profile, title, desc) in zip(axes, profiles):
    prob_original = base.probability(distances)
    prob_transformed = profile.probability(distances)
    
    ax.plot(distances.mantissa, prob_original, '--', linewidth=2, 
            label='Original', alpha=0.5)
    ax.plot(distances.mantissa, prob_transformed, linewidth=2, label='Transformed')
    ax.set_title(f'{title}\n({desc})')
    ax.set_xlabel('Distance (μm)')
    ax.set_ylabel('Probability')
    ax.legend()
    ax.grid(alpha=0.3)

plt.suptitle('Custom Transformations with .apply()', fontsize=14, y=1.00)
plt.tight_layout()
plt.show()

print("\n🔧 Transformation examples:")
print("  x² → Emphasize strong connections")
print("  √x → Flatten differences")
print("  1-x → Create inhibitory pattern")
print("  sigmoid → Binarize connectivity")
../_images/8bdcf95f59e9920a69dea09cf118ed0570c452ab7bf846e9819fd13a59c09545.png 
🔧 Transformation examples:
  x² → Emphasize strong connections
  √x → Flatten differences
  1-x → Create inhibitory pattern
  sigmoid → Binarize connectivity

4.4 Pipe Operator#

The pipe operator (|) chains operations:

result = profile | (lambda x: x * 2) | (lambda x: np.minimum(x, 1.0))

# Create a complex pipeline
base = init.GaussianProfile(sigma=50.0 * u.um)

# Chain operations with pipe operator
pipeline = (
    base 
    | (lambda x: x * 1.5)           # Scale up
    | (lambda x: np.minimum(x, 1.0)) # Clip to max 1
    | (lambda x: x ** 0.8)           # Slight power transform
)

# Alternative: without pipe operator (nested)
manual = base.apply(lambda x: (np.minimum(x * 1.5, 1.0)) ** 0.8)

# Visualize
fig, axes = plt.subplots(1, 2, figsize=(14, 5))

distances = np.linspace(0, 200, 500) * u.um

prob_base = base.probability(distances)
prob_pipeline = pipeline.probability(distances)
prob_manual = manual.probability(distances)

# Pipeline result
axes[0].plot(distances.mantissa, prob_base, '--', linewidth=2, label='Original', alpha=0.5)
axes[0].plot(distances.mantissa, prob_pipeline, linewidth=2, label='Pipeline')
axes[0].set_xlabel('Distance (μm)', fontsize=11)
axes[0].set_ylabel('Probability', fontsize=11)
axes[0].set_title('Pipe Operator: base | scale | clip | power', fontsize=12)
axes[0].legend()
axes[0].grid(alpha=0.3)

# Verify equivalence
axes[1].plot(distances.mantissa, prob_pipeline, linewidth=2, label='Pipeline', alpha=0.7)
axes[1].plot(distances.mantissa, prob_manual, '--', linewidth=2, label='Manual (nested)', alpha=0.7)
axes[1].set_xlabel('Distance (μm)', fontsize=11)
axes[1].set_ylabel('Probability', fontsize=11)
axes[1].set_title('Pipeline vs Manual (should match)', fontsize=12)
axes[1].legend()
axes[1].grid(alpha=0.3)

plt.tight_layout()
plt.show()

print("\n⛓️ Pipe operator benefits:")
print("  ✓ More readable (left-to-right flow)")
print("  ✓ Easier to modify pipeline")
print("  ✓ Composable and modular")
print(f"\n  Pipeline ≈ Manual: {np.allclose(prob_pipeline, prob_manual)}")
../_images/312ce3438cdb977c1733939f8cb236c3f47d565ba988347e6d4a0a33e80af952.png 
⛓️ Pipe operator benefits:
  ✓ More readable (left-to-right flow)
  ✓ Easier to modify pipeline
  ✓ Composable and modular

  Pipeline ≈ Manual: True

Complex Composition Example#

# Build a complex connectivity pattern
# Goal: Strong local, weak intermediate, moderate long-range

local_excitation = init.GaussianProfile(sigma=30.0 * u.um)
intermediate_inhibition = init.GaussianProfile(sigma=80.0 * u.um)
long_range_excitation = init.GaussianProfile(sigma=150.0 * u.um)

# Compose with weights
complex_pattern = (
    local_excitation * 1.0
    - intermediate_inhibition * 0.3
    + long_range_excitation * 0.2
).clip(min_val=0.0, max_val=1.0)

# Visualize
fig = plt.figure(figsize=(14, 10))
gs = GridSpec(3, 2, figure=fig)

distances = np.linspace(0, 300, 500) * u.um

# Components
ax1 = fig.add_subplot(gs[0, :])
ax1.plot(distances.mantissa, local_excitation.probability(distances), 
         label='Local excitation', linewidth=2)
ax1.plot(distances.mantissa, intermediate_inhibition.probability(distances) * 0.3, 
         label='Intermediate inhibition × 0.3', linewidth=2)
ax1.plot(distances.mantissa, long_range_excitation.probability(distances) * 0.2, 
         label='Long-range excitation × 0.2', linewidth=2)
ax1.set_xlabel('Distance (μm)')
ax1.set_ylabel('Component strength')
ax1.set_title('Individual Components')
ax1.legend()
ax1.grid(alpha=0.3)

# Composed pattern
ax2 = fig.add_subplot(gs[1, :])
prob_complex = complex_pattern.probability(distances)
ax2.plot(distances.mantissa, prob_complex, linewidth=3, color='purple')
ax2.fill_between(distances.mantissa, prob_complex, alpha=0.3, color='purple')
ax2.axhline(0, color='black', linestyle='-', linewidth=0.5)
ax2.set_xlabel('Distance (μm)')
ax2.set_ylabel('Connection Probability')
ax2.set_title('Complex Composed Pattern (clipped to [0, 1])')
ax2.grid(alpha=0.3)

# 2D visualization
ax3 = fig.add_subplot(gs[2, 0])
x = np.linspace(-150, 150, 100)
y = np.linspace(-150, 150, 100)
X, Y = np.meshgrid(x, y)
D = np.sqrt(X**2 + Y**2) * u.um
P = complex_pattern.probability(D)
im = ax3.imshow(P, extent=[-150, 150, -150, 150], origin='lower', cmap='RdYlBu_r')
ax3.set_xlabel('X position (μm)')
ax3.set_ylabel('Y position (μm)')
ax3.set_title('2D Spatial Pattern')
plt.colorbar(im, ax=ax3, label='Probability')

# Cross-section
ax4 = fig.add_subplot(gs[2, 1])
center_row = P[50, :]
ax4.plot(x, center_row, linewidth=2)
ax4.fill_between(x, center_row, alpha=0.3)
ax4.set_xlabel('Distance from center (μm)')
ax4.set_ylabel('Connection Probability')
ax4.set_title('Cross-section Through Center')
ax4.grid(alpha=0.3)

plt.suptitle('Complex Multi-Scale Connectivity Pattern', fontsize=14, y=0.995)
plt.tight_layout()
plt.show()

print("\n🧠 This pattern creates:")
print("  1. Strong local excitation (0-50 μm)")
print("  2. Lateral inhibition (50-150 μm)")
print("  3. Weak long-range excitation (150-300 μm)")
print("\n  Common in cortical circuits!")
../_images/b2240654057d6b5ced9d472b22b42ec1faec572327542118ba71b2fddb40134c.png 
🧠 This pattern creates:
  1. Strong local excitation (0-50 μm)
  2. Lateral inhibition (50-150 μm)
  3. Weak long-range excitation (150-300 μm)

  Common in cortical circuits!

5. Real-World Neuroscience Applications#

Application 1: V1 Orientation Columns#

# Model V1 orientation column connectivity
# Same orientation: strong local connection
# Different orientation: lateral inhibition

# Simplified: distance-based approximation
same_orientation = init.GaussianProfile(sigma=50.0 * u.um)
different_orientation = (
    init.DoGProfile(
        sigma_center=40.0 * u.um,
        sigma_surround=100.0 * u.um,
        amplitude_center=0.3,
        amplitude_surround=0.4
    )
).clip(min_val=0.0)

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

distances = np.linspace(0, 300, 500) * u.um

# Same orientation
prob_same = same_orientation.probability(distances)
axes[0].plot(distances.mantissa, prob_same, linewidth=2, color='blue')
axes[0].fill_between(distances.mantissa, prob_same, alpha=0.3, color='blue')
axes[0].set_xlabel('Distance (μm)', fontsize=11)
axes[0].set_ylabel('Connection Probability', fontsize=11)
axes[0].set_title('Same Orientation Preference', fontsize=12)
axes[0].grid(alpha=0.3)

# Different orientation
prob_diff = different_orientation.probability(distances)
axes[1].plot(distances.mantissa, prob_diff, linewidth=2, color='red')
axes[1].fill_between(distances.mantissa, prob_diff, alpha=0.3, color='red')
axes[1].set_xlabel('Distance (μm)', fontsize=11)
axes[1].set_ylabel('Connection Probability', fontsize=11)
axes[1].set_title('Different Orientation Preference', fontsize=12)
axes[1].grid(alpha=0.3)

plt.suptitle('V1 Orientation Column Connectivity', fontsize=14, y=1.02)
plt.tight_layout()
plt.show()

print("\n👁️ V1 orientation columns:")
print("  - Neurons preferring same orientation connect strongly locally")
print("  - Neurons with different orientations show lateral inhibition")
print("  - Creates orientation selectivity sharpening")
../_images/6a1eae37ee4e70a54455b13792c97ec0163f87b7c661cd5bb7d82440f35270ac.png 
👁️ V1 orientation columns:
  - Neurons preferring same orientation connect strongly locally
  - Neurons with different orientations show lateral inhibition
  - Creates orientation selectivity sharpening

Application 2: Hippocampal Place Cells#

# Hippocampal place cell connectivity
# Overlapping place fields → stronger connection

def place_field_overlap(distance, field_size=40.0):
    """
    Model place field overlap as function of cell distance.
    """
    # Simplified: Gaussian overlap
    profile = init.GaussianProfile(sigma=field_size * u.cm)
    return profile.probability(distance)

# CA3 recurrent connections: bimodal (local + long-range)
ca3_recurrent = init.BimodalProfile(
    sigma1=50.0 * u.um,
    sigma2=200.0 * u.um,
    center1=0.0 * u.um,
    center2=300.0 * u.um,
    amplitude1=0.8,
    amplitude2=0.3
)

# CA1 connections: more local
ca1_local = init.GaussianProfile(sigma=80.0 * u.um)

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

distances = np.linspace(0, 500, 500) * u.um

# CA3 recurrent
prob_ca3 = ca3_recurrent.probability(distances)
axes[0].plot(distances.mantissa, prob_ca3, linewidth=2)
axes[0].fill_between(distances.mantissa, prob_ca3, alpha=0.3)
axes[0].set_xlabel('Distance (μm)', fontsize=11)
axes[0].set_ylabel('Connection Probability', fontsize=11)
axes[0].set_title('CA3 Recurrent Connectivity', fontsize=12)
axes[0].grid(alpha=0.3)

# CA1 local
prob_ca1 = ca1_local.probability(distances)
axes[1].plot(distances.mantissa, prob_ca1, linewidth=2, color='orange')
axes[1].fill_between(distances.mantissa, prob_ca1, alpha=0.3, color='orange')
axes[1].set_xlabel('Distance (μm)', fontsize=11)
axes[1].set_ylabel('Connection Probability', fontsize=11)
axes[1].set_title('CA1 Local Connectivity', fontsize=12)
axes[1].grid(alpha=0.3)

plt.suptitle('Hippocampal Connectivity Patterns', fontsize=14, y=1.02)
plt.tight_layout()
plt.show()

print("\n🧠 Hippocampal connectivity:")
print("  CA3: Bimodal (supports pattern completion)")
print("  CA1: More local (different computational role)")
print("  Place cells with overlapping fields connect preferentially")
../_images/530c8c0746c85e1d7e3b2e5393644ebb3638c866aea0788b85746160381e47c5.png 
🧠 Hippocampal connectivity:
  CA3: Bimodal (supports pattern completion)
  CA1: More local (different computational role)
  Place cells with overlapping fields connect preferentially

Application 3: Cortical Layer-Specific Connectivity#

# Different cortical layers have different connectivity patterns

# Layer 2/3: Local horizontal connections
layer23 = init.GaussianProfile(sigma=200.0 * u.um)

# Layer 4: Dense local connections
layer4 = init.GaussianProfile(sigma=100.0 * u.um)

# Layer 5: Long-range projections
layer5 = init.BimodalProfile(
    sigma1=100.0 * u.um,
    sigma2=400.0 * u.um,
    center1=0.0 * u.um,
    center2=600.0 * u.um,
    amplitude1=0.7,
    amplitude2=0.4
)

# Layer 6: Feedback connections
layer6 = init.ExponentialProfile(decay_constant=300.0 * u.um)

# Visualize
fig, axes = plt.subplots(2, 2, figsize=(14, 10))
axes = axes.flatten()

distances = np.linspace(0, 800, 500) * u.um

layers = [
    (layer23, 'Layer 2/3: Horizontal Connections', 'blue'),
    (layer4, 'Layer 4: Dense Local', 'green'),
    (layer5, 'Layer 5: Long-range Projections', 'red'),
    (layer6, 'Layer 6: Feedback', 'purple'),
]

for ax, (profile, title, color) in zip(axes, layers):
    prob = profile.probability(distances)
    ax.plot(distances.mantissa, prob, linewidth=2, color=color)
    ax.fill_between(distances.mantissa, prob, alpha=0.3, color=color)
    ax.set_xlabel('Distance (μm)', fontsize=11)
    ax.set_ylabel('Connection Probability', fontsize=11)
    ax.set_title(title, fontsize=12)
    ax.grid(alpha=0.3)

plt.suptitle('Layer-Specific Cortical Connectivity', fontsize=14, y=1.00)
plt.tight_layout()
plt.show()

print("\n📊 Layer-specific properties:")
print("  L2/3: Broad horizontal connections (integrate information)")
print("  L4: Dense local (receive thalamic input)")
print("  L5: Long-range projections (output layer)")
print("  L6: Feedback to thalamus (modulation)")
../_images/2c79bb956412ebfa34b6eca7f069dc84dff40ea8fb430ff2a788527e484555c4.png 
📊 Layer-specific properties:
  L2/3: Broad horizontal connections (integrate information)
  L4: Dense local (receive thalamic input)
  L5: Long-range projections (output layer)
  L6: Feedback to thalamus (modulation)

Summary#

In this tutorial, we covered:

  1. Basic profiles: Gaussian, Exponential, PowerLaw, Linear, Step

  2. Advanced profiles: Sigmoid, DoG, Logistic, Bimodal, MexicanHat

  3. Profile composition: Arithmetic operations, clipping, transformations

  4. Pipe operator: Functional composition for clean code

  5. Real applications: V1, hippocampus, cortical layers

Profile Selection Guide:#

Pattern

Profile

Local excitation

Gaussian

Long-range decay

Exponential

Scale-free

PowerLaw

Hard boundary

Step, Linear

Smooth transition

Sigmoid, Logistic

Lateral inhibition

DoG, MexicanHat

Multi-scale

Bimodal, Composed