Stochastic Hodgkin-Huxley Model: 14x28D Langevin Simulation (Pu and Thomas, 2020).

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This model provides a natural 14-dimensional Langevin dynamics for the Hodgkin Huxley system in which each directed edge in the ion channel state transition graph acts as an independent noise source, leading to a 14 dimensional state space (1 dimension for voltage, 5 for potassium and 8 for sodium) and 14 × 28 noise coefficient matrix S. In [Pu and Thomas (2020) Neural Computation] we show that this 14 x 28 dimensional model is pathwise equivalent to the 14 x 11 dimensional Langevin model proposed in [Fox and Lu (1994) Phys Rev E], as well as an 14 x 14 model described in [Orio and Soudry (2012) PLoS One]. Unlike Fox and Lu's model, our construction does not require a matrix root extraction step, and runs significantly faster. Unlike Orio and Soudry's model, each directed edge acts as an independent noise source, which facilitates the application of stochastic shielding methods for even greater simulation speed. For comparison, we provide implementations of the following models: 1. Discrete-state Markov chain model (slow, but provides the "gold standard" model), adapted from [Goldwyn and Shea-Brown (2011) PLoS Comp. Biol.] 2. 14 x 11 Langevin model from [Fox and Lu (1994) Phys. Rev. E]. (We implement versions with three different boundary conditions: open boundaries, reflecting boundaries, and resampling/rejection at the boundaries.) 3. 4 x 3 Langevin model from [Fox (1997) Biophys. J.] 4. 14 x 13 Langevin model from [Goldwyn and Shea (2011) PLoS Comp. Biol.] 5. 14 x 14 Langevin model from [Dangerfield et al (2012) Phys. Rev. E] 6. 14 x 14 Langevin model from [Orio and Soudry (2012) PLoS One] 7. 14 x 28 Langevin model from [Pu and Thomas (2020) Neural Computation] implemented both with and without stochastic shielding 8. 14 x 0 deterministic HH model (also from [Pu and Thomas (2020) Neural Computation], with the full 14 dimensional state space but no noise) The file provides more detailed simulations. To cite the code: Pu, Shusen, and Peter J. Thomas. "Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics." Neural Computation 32, 1775–1835 (2020)
1 . Pu S, Thomas PJ (2020) Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics Neural Computation 32:1775-1835
Model Information (Click on a link to find other models with that property)
Model Type: Neuron or other electrically excitable cell;
Brain Region(s)/Organism: Generic;
Cell Type(s): Hodgkin-Huxley neuron;
Channel(s): I K; I Na, leak;
Gap Junctions:
Simulation Environment: MATLAB;
Model Concept(s): Stochastic simulation;
Search NeuronDB for information about:  I K; I Na, leak;
# Matlab code for Pu and Thomas' Neural Computation paper
## Fast and Accurate Langevin Simulations of Stochas-tic Hodgkin-Huxley Dynamics
1. Figure 1 (HH diagram) is generated by the tex code.
2. Figure 2 (4D and 14D HH models) can be generated from codes in folder "Figure2". It takes roughly 3-5 minutes to run the script Fig2.m on a laptop.
3. Figure 3 (convergence to the multinomial submanifold) can be generated by codes in folder "Figure3".  It takes roughly 3-5 minutes to run the script Fig3.m on a laptop.
4. Figure 4 (Edge importance under voltage clamp) is reproduced with permission from Figs. 10 & 13 of [Schmidt and Thmas (2014) paper](  Code is reproduced with permission.  
5. Figure 5 (edge importance under current clamp) can be reproduced from codes in folder "Figure5". 
   - to generate figure 5, one needs to run hundreds repeated simulations, high performance computing is recommended
   - one can modify the code gene_data_SS.m to generate as many samples as desired 
   - fig5data.mat can be loaded and viewed by plot_fig5.m
6. Figure 6 (pathwise equivalency) can be generated by the code Fig6.m, which takes roughly 1-2 minutes to run on a laptop.
7. Figures 7, 8 and 9, as well as codes for generating Table 3, are included in the folder "All_models"
   - cluster computing or high performance computing is recommended for generating the data for these tables and figures
   - details of the simulation for the paper is specified in Section 5
   - simulation efficiency is computed through the same laptop computer; the time might be different using different machines but the ratio should  be roughly the same
   - since the data for all plots in Figs. 7-9 and Table 3 are more than 500 MB, data for the plots are not uploaded here but one should be able to re-generate the data with the provided code

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