## Circuit Prototype

The gain is $V=\frac{kT}{q}=25.4$ mV per e-fold intensity change.

Deficient in 2 respects:

1. High signal offset
2. Slow response

• left side: source follower
• lowpass feedback
• internal adaptive learning model to predict the input signal.
• output: the comparison of the input signal and prediction.

Right side: Comparison model

• $Q_n, Q_{cas}, Q_{p}$ consist a inverting amplifier for feedback.
• $V_b$: determines the cutoff frequency for receptor, by setting the bias current.
• $Q_{cas}$: cascode which nullifies the Miller capacitance
• feedback loop: adaptive element and capacitor $C_1$ and $C_2$

## Circuit Performance

Steady-state closed-loop gain: $$\frac{\frac{v_o}{V_T}}{\frac{i}{l_{bg}}} = \frac{1}{\kappa}$$

Transient closed-loop gain: $$A_{cl} \equiv \frac{\frac{v_o}{V_T}}{\frac{i}{l_{bg}}} = \frac{1}{\kappa} \frac{C_1+C_2}{C_2}$$

Time constant of the receptor:(baseline speed) $$\tau_{in} = \frac{C}{g} = \frac{C V_T}{I_{bg}}$$

The speedup obtained by using the active feedback to clamp the input node is given by: $$A_{loop} = \frac{A_{amp}}{A_{cl}}$$

The total receptor noise in dimensionlessinput-referred units is given by: $$\frac{\Delta i^2}{I_{bg}^{2}} = \frac{q}{C_{eff}V_T}$$ where $C_{eff}$ is the effective input capacitance including the effect of total loop gain

The spectral density of the noise within the passband is given by a simple computa- tion involving the first-order low-pass filter at the input:(noise spectral density) $$S = \frac{4q}{I_{bg}}$$

Second-order temporal effects:

A second-order time-domain analysis reveals that in order for the receptor to be a first-order system (i.e. produce a non-ringing response to a step input), the output-node time constant of the feedback amplifier must be faster than the input node time constant by a factor of $A_{loop}$. However, since the maximum Q of the circuit only scales as $\sqrt{A_{loop}}$, a large bias current may not be essential. We usually tolerate some resonance in return for filtering out flicker from artificial lighting and reducing noise at high intensities.

Dynamic Range:

The usable bandwidth of the receptor is proportional to the intensity. The receptor has a bandwidth of at least 60Hz (like human photopic vision) down to an irradiance of about 1 mW/m^2. This irradiance corresponds to an illuminance of about 1 lux for 555 nm (yellow) photons.

## Receptor Characteristics

• The amplitude of the response to the small contrast variation is almost invariant to the absolute intensity, owing to the logarithmic response property. The adaptation makes the receptor have high gain for rapidly varying intensities and low gain for slowly varying intensities. A careful examination shows that the receptor adapts very rapidly in response to the decade changes in intensity. This rapid adaptation is due to the use of an adaptive element with an expansive nonlinearity.
• Adaptation occurs when charge is transferred through the adaptive element onto or off the storage capacitor.The adaptive element acts like a pair of diodes, in parallel, with opposite polarity. The current increases exponentially with voltage for either sign of voltage, and there is an extremely high-resistance region around the origin, as shown in Figure 4. The I-V relationship of the expansive element means that the effective resistance of the element is huge for small signals and small for large signals. Hence, the adaptation is slow for small signals and fast for large signals.

## Source

DOI: 10.1109/iscas.1994.409266 Ref: DELBRUCK, T. & MEAD, C. A. Adaptive photoreceptor with wide dynamic range. IEEE.