This article is about the property of a stochastic process. zeros_like ( t ) switch_times = # assume we always start in Closed state x = 0 # generate a bunch of random uniformly distributed numbers # between zero and unity: = 0 and myrand < c2o * dt : # remember to scale by dt! x = 1 switch_times. # Execute to simulate and plot state changes # parameters T = 5000 # total Time duration dt = 0.001 # timestep of our simulation # simulate state of our ion channel in time # the two parameters that govern transitions are # c2o: closed to open rate # o2c: open to closed rate def ion_channel_opening ( c2o, o2c, T, dt ): # initialize variables t = np. Commenting out that line will produce a different simulation each run.Įxecute to simulate and plot state changes Note that a random seed was set in the code block, so re-running the code will produce the same plot. Run the cell below to show the state-change simulation process. You briefly saw a Markov process in the pre-reqs statistics day. We can use those times to measure the distribution of the time intervals between state switches. In the simulation below, we will use the Poisson process to model the state of our ion channel at all points \(t\) within the total simulation time \(T\).Īs we simulate the state change process, we also track at which times throughout the simulation the state makes a switch. Our ion channel can either be in an open or closed state, but not both simultaneously. Two events cannot occur at the same moment. The average rate of events within a given time period is constant. The probability of some event occurring is independent from all other events. Importantly, the Poisson process dictates the following points: The Poisson process is a way to model discrete events where the average time between event occurrences is known but the exact time of some event is not known. ![]() You have seen the Poisson process in the pre-reqs statistics day. We simulate the process of changing states as a Poisson process. Put another way, dynamical systems that involve probability will incorporate random variations in their behavior.įor some probabilistic dynamical systems, the differential equations express a relationship between \(\dot\). Every time you observe some probabilistic dynamical system, starting from the same initial conditions, the outcome will likely be different. In a probabilistic process, elements of randomness are involved. You may sometimes hear these systems called stochastic. For Tutorial 2, we will look at probabilistic dynamical systems. In Tutorial 1, we studied dynamical systems as a deterministic process. In this tutorial, we will look at the dynamical systems introduced in the first tutorial through a different lens. Tutorial 3: Simultaneous fitting/regressionĮxample Model Project: the Train Illusion Tutorial 4: Model-Based Reinforcement Learning ![]() Tutorial 2: Learning to Act: Multi-Armed Bandits Tutorial 2: Optimal Control for Continuous State Tutorial 1: Optimal Control for Discrete States Tutorial 1: Sequential Probability Ratio Testīonus Tutorial 4: The Kalman Filter, part 2īonus Tutorial 5: Expectation Maximization for spiking neurons Tutorial 2: Bayesian inference and decisions with continuous hidden state Tutorial 1: Bayes with a binary hidden state Tutorial 3: Synaptic transmission - Models of static and dynamic synapsesīonus Tutorial: Spike-timing dependent plasticity (STDP)īonus Tutorial: Extending the Wilson-Cowan Model Tutorial 1: The Leaky Integrate-and-Fire (LIF) Neuron Model Tutorial 3: Combining determinism and stochasticity Tutorial 3: Building and Evaluating Normative Encoding Modelsīonus Tutorial: Diving Deeper into Decoding & Encoding Tutorial 2: Convolutional Neural Networks Tutorial 4: Nonlinear Dimensionality Reduction Tutorial 3: Dimensionality Reduction & Reconstruction Tutorial 6: Model Selection: Cross-validation Tutorial 5: Model Selection: Bias-variance trade-off ![]() Tutorial 4: Multiple linear regression and polynomial regression Tutorial 3: Confidence intervals and bootstrapping Tutorial 1: Differentiation and Integration Prerequisites and preparatory materials for NMA Computational Neuroscienceīonus Tutorial: Discrete Dynamical Systems
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