Sensory Processing

Derivation 1: Center-Surround Receptive Fields as Difference of Gaussians

The center-surround receptive field can be modeled as a Difference of Gaussians (DoG). For an ON-center cell at position $(x_0, y_0)$:

$$RF(x, y) = \frac{A_c}{2\pi\sigma_c^2} \exp\left(-\frac{(x-x_0)^2 + (y-y_0)^2}{2\sigma_c^2}\right) - \frac{A_s}{2\pi\sigma_s^2} \exp\left(-\frac{(x-x_0)^2 + (y-y_0)^2}{2\sigma_s^2}\right)$$

where $A_c, \sigma_c$ are the center amplitude and width, and $A_s, \sigma_s$ are the surround parameters. Typically $\sigma_s \approx 3\sigma_c$ and the total integral is approximately zero, ensuring the cell responds to contrast rather than absolute luminance.

The response to a stimulus $I(x,y)$ is computed by convolution:

$$r = \int\int RF(x, y) \cdot I(x, y) \, dx \, dy$$

In the Fourier domain, the DoG acts as a bandpass filter: $\widetilde{RF}(f) = A_c e^{-2\pi^2 \sigma_c^2 f^2} - A_s e^{-2\pi^2 \sigma_s^2 f^2}$, which peaks at an optimal spatial frequency $f^* = \frac{1}{2\pi}\sqrt{\frac{2\ln(A_c\sigma_s^2 / A_s\sigma_c^2)}{\sigma_s^2 - \sigma_c^2}}$.

Derivation 2: Gabor Model of V1 Simple Cell Receptive Fields

A V1 simple cell's receptive field is well described by a Gabor function — a sinusoid modulated by a Gaussian envelope. In 2D:

$$G(x, y) = A \exp\left(-\frac{x'^2 + \gamma^2 y'^2}{2\sigma^2}\right) \cos(2\pi f x' + \phi)$$

where the rotated coordinates are:

$$x' = x\cos\theta + y\sin\theta, \quad y' = -x\sin\theta + y\cos\theta$$

Here $\theta$ is the preferred orientation, $f$ is the spatial frequency,$\sigma$ is the Gaussian width, $\gamma$ is the aspect ratio (typically$\gamma \approx 0.5$), and $\phi$ is the spatial phase. The orientation bandwidth (half-width at half-maximum) is:

$$\Delta\theta_{1/2} = \arctan\left(\frac{1}{\pi f \sigma}\sqrt{\frac{\ln 2}{2}}\right)$$

Typical V1 neurons have orientation bandwidths of 20–40 degrees and spatial frequency bandwidths of 1–1.5 octaves. Daugman (1985) showed that Gabor functions achieve the theoretical minimum joint uncertainty in space and spatial frequency, making them optimal local feature detectors.

Derivation 3: Orientation Tuning from Feedforward Connectivity

The feedforward model (Hubel and Wiesel, 1962) proposes that a V1 simple cell's orientation selectivity arises from the alignment of LGN inputs. Consider a simple cell receiving input from $N$ LGN cells with center-surround receptive fields arranged along a line at angle $\theta_0$:

$$RF_{\text{simple}}(x, y) = \sum_{i=1}^{N} w_i \cdot DoG(x - x_i, y - y_i)$$

When centers $(x_i, y_i)$ lie along the preferred axis and weights $w_i$ follow a Gaussian envelope, the resulting receptive field approximates a Gabor function. The orientation tuning curve is approximately Gaussian:

$$r(\theta) = r_{\max} \exp\left(-\frac{(\theta - \theta_0)^2}{2\sigma_\theta^2}\right) + r_{\text{base}}$$

However, the feedforward model alone produces broader tuning than observed. Recurrent cortical interactions sharpen tuning through a combination of recurrent excitation among similarly tuned neurons and cross-orientation inhibition, producing the narrow tuning widths ($\sigma_\theta \approx 15\text{--}25^\circ$) observed experimentally.

Derivation 4: Population Vector Decoding of Orientation

For a population of $N$ neurons with Gaussian orientation tuning curves and independent Poisson noise, the maximum likelihood estimate of the stimulus orientation$\hat{\theta}$ can be obtained analytically. The log-likelihood is:

$$\ln P(\mathbf{r} | \theta) = \sum_{i=1}^{N} \left[ r_i \ln f_i(\theta) - f_i(\theta) - \ln(r_i!) \right]$$

Setting the derivative to zero:

$$\frac{\partial}{\partial\theta} \ln P = \sum_{i=1}^{N} \left(\frac{r_i}{f_i(\theta)} - 1\right) f_i'(\theta) = 0$$

For uniformly spaced preferred orientations and narrow tuning, this simplifies to a population vector:

$$\hat{\theta} = \frac{1}{2}\arg\left(\sum_{i=1}^{N} r_i \, e^{2i\theta_i^{\text{pref}}}\right)$$

The factor of 2 in the exponent accounts for the 180-degree periodicity of orientation. The variance of this estimate satisfies the Cramer-Rao bound:$\text{Var}(\hat{\theta}) \geq 1/J(\theta)$ where $J(\theta) = \sum_i [f_i'(\theta)]^2 / f_i(\theta)$scales linearly with $N$.

Derivation 5: Noise Correlations and Population Information

In reality, neurons share noise correlations $\rho_{ij}$, which fundamentally alter the information content of the population. The Fisher information for correlated neurons is:

$$J(\theta) = \mathbf{f}'(\theta)^T \, \mathbf{Q}^{-1} \, \mathbf{f}'(\theta) + \frac{1}{2}\text{tr}\left[\mathbf{Q}^{-1}\frac{\partial\mathbf{Q}}{\partial\theta}\mathbf{Q}^{-1}\frac{\partial\mathbf{Q}}{\partial\theta}\right]$$

where $\mathbf{Q}$ is the noise covariance matrix and $\mathbf{f}'(\theta)$ is the vector of tuning curve derivatives. For a population with uniform pairwise correlation$\rho$ between all neuron pairs:

$$J_N(\theta) \approx \frac{J_1(\theta)}{1 + (N-1)\rho} \cdot N$$

When $\rho > 0$ (positive correlations, as commonly observed), information saturates at $J_\infty = J_1 / \rho$ as $N \to \infty$. This means that adding more neurons cannot improve precision beyond a limit set by correlation structure — a result with profound implications for neural coding efficiency (Zohary et al., 1994; Averbeck et al., 2006).