The code provided is for computing the cumulative distribution function (CDF) of the von Mises distribution. The von Mises distribution is a continuous probability distribution on the circle, often used in the analysis of circular data. In the context of computational neuroscience, this distribution can model phenomena where directionality or phase is essential, and quantities are represented as angles (e.g., on a unit circle). ### Biological Basis 1. **Circular Statistics in Neuroscience:** - Many neural phenomena are inherently directional or cyclical, such as the orientation preference of neurons in the visual cortex, the directional tuning of head-direction cells, and the phase locking of neurons to oscillatory patterns. In these cases, data are more naturally represented in circular or angular forms. 2. **Von Mises Distribution:** - The von Mises distribution serves as an analogue to the normal distribution but on a circular domain. It is characterized by a mean direction (`mu`) and a concentration parameter (`kappa`), which indicate the preferred direction and the strength of clustering around this direction, respectively. - In biological terms, `mu` could represent the preferred direction or phase of a neural property, while `kappa` describes how tightly the data is concentrated around that preferred direction, analogous to precision or reliability of neuronal response. 3. **Applications in Neural Systems:** - **Orientation Selectivity:** Neurons in the visual cortex exhibit orientation selectivity, responding preferentially to edges of a specific angle. The von Mises distribution can model the probability distribution of orientation responses. - **Hippocampal Place Cells and Grid Cells:** These cells exhibit spatial firing patterns that are periodic and directional. The von Mises distribution might be used to describe the firing probability around specific directions or phases of a spatial input. - **Head-Direction Cells:** These cells encode the animal’s head orientation in the environment. The von Mises CDF could be used to predict the cumulative response probability corresponding to a range of head directions. The code, through the modeling of the von Mises CDF, captures the probabilistic nature of these directional neural properties, helping to predict or understand the cumulative response as a function of directional input in a biological system.