From 0282676725dcdaa9a1093a5a7fcfe76bf2d2189a Mon Sep 17 00:00:00 2001 From: chadHarper <116227903+chadHarper@users.noreply.github.com> Date: Tue, 12 Nov 2024 00:59:52 -0800 Subject: [PATCH] Update path-integrals-sdes-neuroscience.md --- _writings/path-integrals-sdes-neuroscience.md | 22 +++++++++---------- 1 file changed, 11 insertions(+), 11 deletions(-) diff --git a/_writings/path-integrals-sdes-neuroscience.md b/_writings/path-integrals-sdes-neuroscience.md index 6e7fadde1..14a4ba523 100644 --- a/_writings/path-integrals-sdes-neuroscience.md +++ b/_writings/path-integrals-sdes-neuroscience.md @@ -33,20 +33,20 @@ InĀ [^4], network population dynamics were modeled as a stochastic hybrid system A **master equation** describes the time evolution of the probability distribution over a set of discrete states in a stochastic system. A general form is: -\[ +$$ \frac{dP_i(t)}{dt} = \sum_{j \neq i} \left[ R_{j \to i} P_j(t) - R_{i \to j} P_i(t) \right], -\] +$$ where: -- \( P_i(t) \) is the probability of the system being in state \( i \) at time \( t \). -- \( R_{j \to i} \) is the transition rate from state \( j \) to state \( i \). +- $ P_i(t) $ is the probability of the system being in state $ i $ at time $ t $. +- $ R_{j \to i} $ is the transition rate from state $ j $ to state $ i $. -This equation captures the balance of probability flow between different states: the first term represents the inflow to state \( i \), while the second term represents the outflow from state \( i \). +This equation captures the balance of probability flow between different states: the first term represents the inflow to state $ i $, while the second term represents the outflow from state $i $. To solve the master equation, one typically seeks: -- **Stationary distributions** where \( \frac{dP_i(t)}{dt} = 0 \) for all \( i \). +- **Stationary distributions** where $ \frac{dP_i(t)}{dt} = 0 $ for all $ i $. - **Time-dependent solutions** to understand transient dynamics, using methods like matrix exponentiation, generating functions, or numerical simulations. Below, we describe the Differential Chapman-Kolmogorov equation (CKdE), which generalizes the master equation and encompasses the Fokker-Planck equation as a special case. @@ -57,17 +57,17 @@ The CKdE describes the dynamics of a stochastic process over time. Here's an out 1. **Start with the Chapman-Kolmogorov Equation**: - For a Markov process with transition probabilities \( p(\mathbf{x}, t | \mathbf{x}_0, t_0) \): + For a Markov process with transition probabilities $p(\mathbf{x}, t | \mathbf{x}_0, t_0)$: - \[ + $$ p(\mathbf{x}, t+\Delta t | \mathbf{x}_0, t_0) = \int_{\Omega} p(\mathbf{x}, t+\Delta t | \mathbf{z}, t) p(\mathbf{z}, t | \mathbf{x}_0, t_0) \, d\mathbf{z}. - \] + $$ 2. **Consider the Time Derivative of the Probability Density**: - \[ + $$ \frac{\partial p(\mathbf{x}, t)}{\partial t} = \lim_{\Delta t \to 0} \frac{1}{\Delta t} \left[ p(\mathbf{x}, t+\Delta t) - p(\mathbf{x}, t) \right]. - \] + $$ 3. **Use the Chapman-Kolmogorov Equation**: