Hi All,

I thought about this project for quite a while, but funnaly I implemented the lookup-table approximation in Brian2.

### What is Lookup table approximation?

The lookup-table approximation is a classical method for computation acceleration in a numerical problem, known for centuries. It is extensively used in such software as NEURON and GENESIS, but I kind of miss it in Brian. This approximation is based on a straightforward algorithm: First, before simulation, one needs to precompute lookup tables for values of m_\infty(v), h_\infty(v), and so on in a full range of voltages. Usually, this range goes between the lowest possible to the highest possible voltages. The range is divided into intervals with constant steps. For example, this may be a range from â100 to 60 mV with 1 mV step. With precomputed tables, one solves differential equations using linear interpolation between table columns instead of computing exponential functions. The voltage at a current time moment of a numerical solution is used to find indices of two columns in the lookup table closest to the membrane voltage. Using these two indices, one can query values for all steady-states and time constants of gating variables and linearly interpolate between these values - like this:

### How to use it in Brian2?

You need to install the module `lut4brian`

first.

```
pip install lut4brian --user
```

In the code one needs to import the module and set up 4 critical module variables:

- lut4brian.Xmin - minimal value of index variable (voltage in the case of ion channels)
- lut4brian.Xmax - maximal value of the same variable
- lut4brian.dX - and the step
- lut4brian.tbl - is a
`numpy`

array with all your values. Rows are different variables (m_\infty(v), h_\infty(v), etc.). Columns are values for the index.

Here is an example of the code:

```
from brian2 import *
from lut4brian import *
x=linspace(-100,50,151)
lut4brian.Xmin = -100.
lut4brian.Xmax = 50.
lut4brian.dX = 1.
alpha_m = 0.1*10./exprel(-(x+35.)/(10.))
beta_m = 4*exp(-(x+60.)/(18.))
minf = alpha_m/(alpha_m+beta_m)
alpha_h = 0.07*exp(-(x+58.)/(20.))
beta_h = 1./(exp(-0.1*(x+28.))+1.)
htau = 1./(alpha_h+beta_h)
hinf = alpha_h*htau
alpha_n = 0.01*10/exprel(-(x+34.)/(10.))
beta_n = 0.125*exp(-(x+44.)/(80.))
ntau = 1./(alpha_n+beta_n)
ninf = alpha_n*ntau
lut4brian.tbl = array([minf,hinf,htau,ninf,ntau])
```

The table `lut4brian.tbl`

has shape, in this case, `(5, 151)`

- 5 variables tabled for 151 points each

For `cpp_standalone`

and `cython`

targets, one *needs to create the table on the level of C-code/Cython* - simple, just call

```
lutinit()
```

this will create `lut4brian.c`

in your current directory. After that tables can be deleted to release memory.

Now, in equation section one can pool interpolation values right out of the table:

```
neurons = NeuronGroup(1000,
'''
dv/dt = (-gNa*minf**3*h*(v-ENa)-gK*n**4*(v-EK)-gL*(v-EL)+I+b*gsyn*(Esyn-v))/Cm : volt
minf = lutinterpol(v/mV,0) : 1
hinf = lutinterpol(v/mV,1) : 1
htau = lutinterpol(v/mV,2) : 1
ninf = lutinterpol(v/mV,3) : 1
ntau = lutinterpol(v/mV,4) : 1
dh/dt = 5*(hinf-h)/htau/ms : 1
dn/dt = 5*(ninf-n)/ntau/ms : 1
db/dt = -b/taus : 1
I : amp
''' , threshold="v>0.*mV", refractory="v>0.*mV", method='euler')
```

The `lutinterpol`

function uses voltage to compute index in the table and returns interpolation value. Note we added individual tables in this order `minf`

, `hinf`

, `htau`

, `ninf`

, `ntau`

and fetched variables by corresponding âcurve indexâ - the second argument of the `lutinterpol()`

function.

### Does it work faster than direct computations?

Well, check it out. In test directory on the GitHub there are a few benchmarks.

Original | Lookup table | |
---|---|---|

Cython | 22 s | 15s |

CPP standalone (single core) | 53s | 21s |

*I havenât checked numpy implementation yet*

I actually expected 3-4 fold acceleration, but on average, it is only twice faster than simulations with âfully computedâ steady-states and taus.

Let me know what do you think.

RTH