Update path-integrals-sdes-neuroscience.md

_writings/path-integrals-sdes-neuroscience.md

@@ -33,20 +33,20 @@ In [^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**: