tweak to img folder for sindy

Author: Alexander Ororbia

docs/museum/sindy.md

@@ -9,7 +9,7 @@ Inconsistent color schemes across visualizations
 -->
 
 
-# Sparse Identification of Non-linear Dynamical Systems (SINDy)[1]
+# Sparse Identification of Non-linear Dynamical Systems (SINDy) 
 
 In this section, we will study, create, simulate, and visualize a model known as the sparse identification of non-linear dynamical systems (SINDy) [1], implementing it in NGC-Learn and JAX. After going through this demonstration, you will:
 

docs/tutorials/neurocog/hodgkin_huxley_cell.md

@@ -83,9 +83,9 @@ Formally, the core dynamics of the H-H cell can be written out as follows:
 
 $$
 \tau_v \frac{\partial \mathbf{v}_t}{\partial t} &= \mathbf{j}_t - g_Na * \mathbf{m}^3_t * \mathbf{h}_t * (\mathbf{v}_t - v_Na) - g_K * \mathbf{n}^4_t * (\mathbf{v}_t - v_K) - g_L * (\mathbf{v}_t - v_L) \\
-\frac{\partial \mathbf{n}_t}{\partial t} &= alpha_n(\mathbf{v}_t) * (1 - \mathbf{n}_t) - beta_n(\mathbf{v}_t) * \mathbf{n}_t \\
-\frac{\partial \mathbf{m}_t}{\partial t} &= alpha_m(\mathbf{v}_t) * (1 - \mathbf{m}_t) - beta_m(\mathbf{v}_t) * \mathbf{m}_t \\
-\frac{\partial \mathbf{h}_t}{\partial t} &= alpha_h(\mathbf{v}_t) * (1 - \mathbf{h}_t) - beta_h(\mathbf{v}_t) * \mathbf{h}_t
+\frac{\partial \mathbf{n}_t}{\partial t} &= \alpha_n(\mathbf{v}_t) * (1 - \mathbf{n}_t) - \beta_n(\mathbf{v}_t) * \mathbf{n}_t \\
+\frac{\partial \mathbf{m}_t}{\partial t} &= \alpha_m(\mathbf{v}_t) * (1 - \mathbf{m}_t) - \beta_m(\mathbf{v}_t) * \mathbf{m}_t \\
+\frac{\partial \mathbf{h}_t}{\partial t} &= \alpha_h(\mathbf{v}_t) * (1 - \mathbf{h}_t) - \beta_h(\mathbf{v}_t) * \mathbf{h}_t
 $$
 
 where we observe that the above four-dimensional set of dynamics is composed of nonlinear ODEs. Notice that, in each gate or channel probability ODE, there are two generator functions (each of which is a function of the membrane potential $\mathbf{v}_t$) that produces the necessary dynamic coefficients at time $t$; $\alpha_x(\mathbf{v}_t)$ and $\beta_x(\mathbf{v}_t)$ produce different biopphysical weighting values depending on which channel $x = \{n, m, h\}$ they are related to.