Simulating neurons or how to solve delay differential equations in R

I discussed earlier how the action potential of a neuron can be modelled via the Hodgkin-Huxely equations. Here I will present a simple model that describes how action potentials can be generated and propagated across neurons. The tricky bit here is that I use delay differential equations (DDE) to take into account the propagation time of the signal across the network.

My model is based on the paper: Epileptiform activity in a neocortical network: a mathematical model by F. Giannakopoulos, U. Bihler, C. Hauptmann and H. J. Luhmann. The article presents a flexible and efficient modelling framework for:

• large populations with arbitrary geometry
• different synaptic connections with individual dynamic characteristics
• cell specific axonal dynamics

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Hodgkin-Huxley model in R

One of the great research papers of the 20th century celebrates its 60th anniversary in a few weeks time: A quantitative description of membrane current and its application to conduction and excitation in nerve by Alan Hodgkin and Andrew Huxley. Only a shortly after Andrew Huxley died, 30th May 2012, aged 94.

In 1952 Hodgkin and Huxley published a series of papers, describing the basic processes underlying the nervous mechanisms of control and the communication between nerve cells, for which they received the Nobel prize in physiology and medicine, together with John Eccles in 1963.

Their research was based on electrophysiological experiments carried out in the late 1940s and early 1950 on a giant squid axon to understand how action potentials in neurons are initiated and propagated.

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Dynamical systems in R with simecol

This evening I will talk about Dynamical systems in R with simecol at the LondonR meeting.

Thanks to the work by Thomas Petzoldt, Karsten Rinke, Karline Soetaert and R. Woodrow Setzer it is really straight forward to model and analyse dynamical systems in R with their deSolve and simecol packages.

I will give a brief overview of the functionality using a predator-prey model as an example.

This is of course a repeat of my presentation given at the Köln R user group meeting in March.

For a further example of a dynamical system with simecol see my post about the Hodgkin-Huxley model, which describes the action potential of a giant squid axon.

I shouldn't forget to mention the other talks tonight as well:

• Writing R for Dummies - Andrie De Vries
• News from data.table 1.6, 1.7 and 1.8 - Matthew Dowle
• Converting S Plus Applications into R - Andy Nicholls (postponed to 18 September 2012)
  

For more information about venue and timing see the LondonR web site.

Logistic map: Feigenbaum diagram in R

The other day I found some old basic code I had written about 15 years ago on a Mac Classic II to plot the Feigenbaum diagram for the logistic map. I remember, it took the little computer the whole night to produce the bifurcation chart.

With today's computers even a for-loop in a scripting language like R takes only a few seconds.

logistic.map <- function(r, x, N, M){
  ## r: bifurcation parameter
  ## x: initial value
  ## N: number of iteration
  ## M: number of iteration points to be returned
  z <- 1:N
  z[1] <- x
  for(i in c(1:(N-1))){
    z[i+1] <- r *z[i]  * (1 - z[i])
  }
  ## Return the last M iterations 
  z[c((N-M):N)]
}

## Set scanning range for bifurcation parameter r
my.r <- seq(2.5, 4, by=0.003)
system.time(Orbit <- sapply(my.r, logistic.map,  x=0.1, N=1000, M=300))
##   user  system elapsed (on a 2.4GHz Core2Duo)
##   2.910   0.018   2.919 

Orbit <- as.vector(Orbit)
r <- sort(rep(my.r, 301))

plot(Orbit ~ r, pch=".")

Let's not forget when Mitchell Feigenbaum started this work in 1975 he did this on his little calculator!

Update, 18 March 2012

The comment from Berend has helped to speedup the code by a factor of about four, thanks to byte compiling (using the same parameters as above), and Owe got me thinking about the alpha value of the plotting colour. Here is the updated result, with the R code below:

library(compiler) ## requires R >= 2.13.0
logistic.map <- cmpfun(logistic.map) # same function as above
my.r <- seq(2.5, 4, by=0.001)
N <- 2000; M <- 500; start.x <- 0.1
orbit <- sapply(my.r, logistic.map,  x=start.x, N=N, M=M)
Orbit <- as.vector(orbit)
r <- sort(rep(my.r, (M+1)))
plot(Orbit ~ r, pch=".", col=rgb(0,0,0,0.05))