LIF neuron integration with delta pulse

[jerem...@gmail.com]

Dear,

I have a very basic question about how a leaky IF neuron is integrated if an incoming spike is considered as a delta function

If I consider a LIF of the form:
dV/dt = (-V + I(t))/tau

I can integrate it exactly and get (with V(0) = 0):
V(t) = 1/tau int_0^t  exp(-s/tau) I(t-s) ds

If I consider an input with a spike arriving at t=t_i, of amplitude g and I take it to be a pulse : 
I(t) = g * delta(t - t_i)
V(t) = g/tau H(t-t_i) exp(-(t-t_i)/tau)
with H(.) the Heaviside function ;

Therefore my increment is of amplitude g/tau ;  while simulating it with Brian with a Connection of weight g gives me an increment of g ;

I make a mistake somewhere but I don't know where;

Thanks for your help;

Jeremy.

[Marcel Stimberg]

Hi Jeremy,

> I have a very basic question about how a leaky IF neuron is integrated
> if an incoming spike is considered as a delta function
>
> If I consider a LIF of the form:
> dV/dt = (-V + I(t))/tau

> [...]

> Therefore my increment is of amplitude g/tau ; while simulating it with
> Brian with a Connection of weight g gives me an increment of g ;

Does your Brian model equation look like 'dV/dt = -V / tau' or are you
explicitly representing the I variable?

The connection does nothing more than add a weight to the target
variable, so if your connection is targetting 'V' (if you do not give an
explicit target, it will also target 'V'), it will indeed increase V by
g and not by g/tau. Hence, the equivalent to modelling I(t) as a delta
function in your formulation is to set the weight in the connection to
g/tau.

Best,
Marcel

[jerem...@gmail.com]

Hi Marcel,

you are true, I was targetting V directly ;

But what if I want to model the Input directly ? so with dV/dt = (-V + I)/tau
If I connect the neurons together they will by default increment "I" which is not good as "I" should just be incremented at one time; Is there a delta synapse ?

jeremy.

[Marcel Stimberg]

Hi Jeremy,

> But what if I want to model the Input directly ? so with dV/dt = (-V +
> I)/tau

Well, there is no direct way to model this in Brian when targetting I --
you could model I as "dI/dt = -I /tau_I" with a very short time constant
tau_I but that would obviously be only an approximation to a delta
synapse. There are other, more complicated solutions, but frankly: Why
would you want to make it more complicated than necessary? For delta
synapses there is no need to have an explicit variable, just directly
target V as you did (with the appropriate weight).

Best,
Marcel

[Jeremy Fix]

It was just for fun to match more closely to the maths of an input
spike train with delta pulses. In math, I would write it as

tau dV/dt = -(V-Vs) + I(t)
I(t) = \sum_i g \delta(t-t_i)

But ok, I will use the workaround.

Thanks,

Best;

jeremy.

2013/1/30 Marcel Stimberg <marcel....@gmail.com>:

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