This repository contains solutions to exercises assigned in the course Physical Models for Living Systems during the academic year 2023-2024. The exercises were tackled in Autumn 2023 and cover topics in ecology modeling (Homework 1-3) and theoretical neuroscience (Homework 4-7). The specific deliveries include:

Homework 1-3: Focus on ecology modeling, covering topics such as linear stability analysis, quasi-stationary approximation, and matrix eigenvalue calculations for different ecological structures.

Homework 4-7: Address theoretical neuroscience, involving tasks such as generating spike trains for neurons, stability analysis of the Wilson-Cowan model, and simulations related to feedback loops in genetic networks.

Deliveries:

1. Perform linear stability analysis of the deterministic logistic equation.
2. Perform the quasi-stationary approximation of the consumer resource model with 1 species and 1 resource for the abiotic case. Check the analytical result with simulations.
3. Simulate the stochastic logistic model on your computer and check that the stationary distribution is a $\Gamma$ (in the case of linear (environmental) multiplicative noise) or $P(n)$ given by the equation for the sub-linear (demographic) multiplicative noise.

Plot both SAR and EAR as a function of the relative area, using the Log Series as the SAD distribution $P(n) = c (1-m)^n / n$ ($c$ is the normalization constant) and compare it with a Power Law SAR of exponent $z=0.25$ ($S(a)=k A^z$) and $k$ tuned so the be as close as possible to the solution for the random placement case.

1. Choose a given type of ecological structure (e.g., mutualistic, or predator-prey, or etc.). Generate random matrices for completely random and for the chosen ecological structures. Create $SxS$ matrices ($S$ is the number of species) with $C$ non-zero entries and $1-C$ zeros (C is the connectivity between 0 and 1). The non-zero elements are drawn at random from given distributions. Depending on the network structure, some symmetries and constraints may hold. Please follow the detailed step-by-step explanation in the "Homework-detail-week3-from-Allesina-Stability-Criteria-2012-Nature.pdf". Fix $C$, and for $S=20, 50, 100, 200$ calculate the eigenvalues of the matrix and plot them against the Real part (x-axis) and imaginary part (y-axis).
2. For case 2) plot the maximum real part eigenvalues for each $S$ (Max $Re \lambda$ as a function of $S$) and the probability of $P(\lambda >0)$ as a function of the proper control parameter (e.g., for the complete random case is $sigma*(S*C)^0.5$). Compare these plots with the one for the complete random case (the original May case).
3. Optional: how are the eigenvalues of the Jacobian of a GLV with constant growth rate set to 1 and random positive $x^*$ (for example taken from a Uniform distribution between 1 and 10) and random interactions distributed? Does the circular or elliptic law still hold? And what about the stability-complexity paradox?

Using non-homogeneous Poisson process, generate the spike train of 100 neurons (if your code is efficient you can also do more, e.g., 200 or 1000).

Do this for two different situations:

1. The firing rate $λ_t$ is independent for each neuron, and generated as random variables extracted each time step by an exponential distribution $p(λ_t) = r exp [- r λ_t]$ with $r=0.1$.
2. The firing rate $λ_t$is the same for each neuron, and generated as random variables extracted each time step by an exponential distribution $p(λ_t) = r exp [- r λ_t]$ with $r=0.1$.

For both cases, you can use a time step $dt=0.01$. Remember that the probability of a spike is $λ_t * dt$. Do at least 1000 time steps.

What are the differences in terms of neural avalanches for the two cases? You can, if you want, characterize the avalanche size distributions, remembering that an avalanche size is defined as the total number of neurons spiking between two periods of total silence (no neurons spiking). Optional: compute the avalanche durations analytically using the same parameters of the simulations for the two cases. The full calculations are available in the notes.

Perform the stability analysis for the Wilson-Cowan model without refractory dynamics. Set self-excitation and self-inhibition to zero, and the $\alpha_E=\alpha_I=1$. Optional: what is the difference if one considers also the refractory dynamics?

Play with the Hopfield network code and/or the Jansen and Rit model code. Understand the code, and play with it. Generate some output (e.g., retrieve the memory input from the Hopfield network). As an output, you can simply send me a doc/pdf file writing a couple of paragraphs, explaining what you have done, and what you have learned. It is enough half a page.

Choose one of the examples of feedback loops (can be either negative or positive) and simulate it. Try to combine two feedback loops (or simulate the genetic network shown in class or creating one yourself), simulate it, and describe in a few sentences the results of the simulations (like "what's going on" in the simulation).