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Tyler Robbins edited MethodsThe_Hindmarsh_Rose_ModelThe_Hindmarsh__.html  over 8 years ago

Commit id: 1bf6c303103eae57a56084d09969203192fefdda

deletions | additions      

       

Hindmarsh-Rose model is a simple model of neuronal activity that allows  for complex behaviors such as bursting and chaotic spiking (Hindmarsh   and Rose 1984). It is described by three coupled first order   differential equations with eight parameters.

x=y-ax^3+bx^2-z+I


y=c-dx^2



z=r(s(x-x_0)-z)`



All parameters.

x=y-ax^3+bx^2-z+I

y=c-dx^2

z=r(s(x-x_0)-z)




All  variables are dimensionless. In these equations, x is the membrane   potential, $y$ is the recovery variable, and z is the slow adaptation   current. The parameters a, b, c, and d model the ion channels that  

and d determining spike frequency. The parameter r forces z to adjust   slowly relative to x and y, while s controls the tendency to burst and   x0 is the resting potential. Finally, I is an applied current from   outside the neuron such as from a patch clamp.



The 




The  HR-neuron model exhibits chaotic characteristics that allow for multiple possible variations on spike times,  shape, and quantity. These chaotic dynamics are well studied  for many possible parameter combinations (Storace,  Linaro, & de Lange, 2008; Shilnikov & Kolomeites, 2008). Researchers have previously 

replacing the applied current, I, with an estimated response function,

x=y-ax^3+bx^2-z+\hat{r}(t)

\hat{r}(t) = h(t) \star s(t)


where  h(t) is a linear filter, s(t) is the stimulus, and $\star$ is   convolution. This adjustment allows us to fit the HR model to in vivio   neuron recording data.




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Genetic class="ltx_title_subsection">

Genetic  Algorithm



Mathematical  models such as the HR-neuron provide researchers a framework to understand and  predict qualities of a given system of interest. Researchers have developed 

the posterior distribution, the ensemble sampler has hundreds of small   chains sampling at once. These features should allow the sampler to   converge quicker than standard MCMC algorithms on good parameter   estimations


Evaluating Fitness


**Tyler - This is where we'll talk about SPIKy and stuff. I'm working on this part now.**



SYNC = \frac{1}{M}\sum_{k=1}^{M}C_k
\[C_i^{(n,m)} = \left \{\begin{matrix}


1 if min_j(|t_i^m-t_j^m)|) <\tau_{i,j}& \\
0 > otherwise &
\end{matrix}\]



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Twin class="ltx_title_subsection">


Twin  Experiments