Unifying Time to Contact Estimation and Collision Avoidance across Species

The -function and the -function are phenomenological models that are widely used in the context of timing interceptive actions and collision avoidance, respectively. Both models were previously considered to be unrelated to each other: is a decreasing function that provides an estimation of time-to-contact (ttc) in the early phase of an object approach; in contrast, has a maximum before ttc. Furthermore, it is not clear how both functions could be implemented at the neuronal level in a biophysically plausible fashion. Here we propose a new framework – the corrected modified Tau function – capable of predicting both -type (“”) and -type (“”) responses. The outstanding property of our new framework is its resilience to noise. We show that can be derived from a firing rate equation, and, as , serves to describe the response curves of collision sensitive neurons. Furthermore, we show that predicts the psychophysical performance of subjects determining ttc. Our new framework is thus validated successfully against published and novel experimental data. Within the framework, links between -type and -type neurons are established. Therefore, it could possibly serve as a model for explaining the co-occurrence of such neurons in the brain.


S8 First Order Temporal Low-Pass Filter (Equation 4)
This section derives equation (4) from the differential equation of a leaky integrator. Let x be the state variable of the leaky integrator which, for example, may represent the firing rate of a neuron: The last equation integrates the input z(t) with time constant τ m (which may represent the membrane time constant of a neuron). If τ m is sufficiently small, then past inputs are quickly discarded, and the filter response x(t) ("output variable") eventually follows the input z(t). In other words, few low-pass filtering of z(t) occurs, and the filter is said to have a short memory. If τ m is very big, then the opposite will occur: The filter gets very sluggish, and eventually sums up all inputs z(t). This means that the filter output x(t) is a strongly low-pass filtered version of the input z(t), and the filter is said to have a long memory. The just described behavior is readily seen when we consider a discretized version of the last equation.
For discretization, we assume that time t increases in steps of ∆t ("sampling interval" or "integration time step"). We have two possibilities for implementing discretizaton: Forward differencing and backward differencing. Both differencing schemes will be considered in turn.

S8.1 Forward Differencing ("Forward Euler")
Here, the right hand side depends on t (i.e., only on past terms): By Rearranging terms we obtain: Now let ξ ≡ ∆t/τ m , and 0 ≤ ξ ≤ 1. Then, we readily obtain equation (4) by defining the memory constants as ζ i ≡ 1 − ξ with i = 1, 2. A big time constant τ m ∆t implies ζ i → 1. This would endow the filter with an infinite memory -it will never change its initial value, because the input z(t) will be multiplied by zero. The other limit case is defined by τ m = ∆t ("small τ m "), and thus ζ i = 0. Then, x(t + ∆t) = z(t), meaning that the filter has no memory on past inputs. In other words, no lowpass filtering takes placethe filter output x follows the input signal z.

S8.2 Backward Differencing ("Backward Euler")
Here, the right hand side depends on t + ∆t (i.e., on future terms): A more compact notation can be obtained by susbtitutingt ≡ t + ∆t in the last equation (and omit the tilde in what follows): By Rearranging terms we obtain: For backward differencing, the filter memory constants ζ i (i = 1, 2) from equation (4) are defined by ζ i ≡ τ m /(τ m + ∆t). Notice that 1 − ζ i = ∆t/(τ m + ∆t), which is the factor associated with the input z. For big time constants τ m ∆t we get ζ i → 1, meaning that our filter would approach an infinite memory (strong lowpass filtering).
For small values τ m = 0, we obtain ζ i = 0 and thus x(t) = z(t) -the filter has no memory on past inputs, and consequently no lowpass filtering will take place.