How to derive the time dependent Leaky Integrator and Fire?

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The general time-dependent LIF model solution is:
$$
u(t) = u_textrest exp left(-fract-t_0tau_mright) + fracRtau_mint_0^t-t_0 expleft(-fracstau_mright) , I(t-s) ,mathrm d s
$$
From the LIF equation
$$
tau_m fracdudt = R I(t) - [u(t) - u_textrest]
$$
According to this paper, equation 7.



It is assumed $t_0$ is the beginning of the time interval, at which $u(t)$ is at it's minimum and that $I(t)$ can vary. I have no idea where to even start, thank you.







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  • Could you please clarify what exactly are you asking? Do you want to understand how to solve LIF equation?
    – Evgeny
    Aug 20 at 9:52











  • Yes, or a step by step derivation of the general LIF so that I can understand it.
    – John Miller
    Aug 20 at 12:01














up vote
0
down vote

favorite
1












The general time-dependent LIF model solution is:
$$
u(t) = u_textrest exp left(-fract-t_0tau_mright) + fracRtau_mint_0^t-t_0 expleft(-fracstau_mright) , I(t-s) ,mathrm d s
$$
From the LIF equation
$$
tau_m fracdudt = R I(t) - [u(t) - u_textrest]
$$
According to this paper, equation 7.



It is assumed $t_0$ is the beginning of the time interval, at which $u(t)$ is at it's minimum and that $I(t)$ can vary. I have no idea where to even start, thank you.







share|cite|improve this question






















  • Could you please clarify what exactly are you asking? Do you want to understand how to solve LIF equation?
    – Evgeny
    Aug 20 at 9:52











  • Yes, or a step by step derivation of the general LIF so that I can understand it.
    – John Miller
    Aug 20 at 12:01












up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





The general time-dependent LIF model solution is:
$$
u(t) = u_textrest exp left(-fract-t_0tau_mright) + fracRtau_mint_0^t-t_0 expleft(-fracstau_mright) , I(t-s) ,mathrm d s
$$
From the LIF equation
$$
tau_m fracdudt = R I(t) - [u(t) - u_textrest]
$$
According to this paper, equation 7.



It is assumed $t_0$ is the beginning of the time interval, at which $u(t)$ is at it's minimum and that $I(t)$ can vary. I have no idea where to even start, thank you.







share|cite|improve this question














The general time-dependent LIF model solution is:
$$
u(t) = u_textrest exp left(-fract-t_0tau_mright) + fracRtau_mint_0^t-t_0 expleft(-fracstau_mright) , I(t-s) ,mathrm d s
$$
From the LIF equation
$$
tau_m fracdudt = R I(t) - [u(t) - u_textrest]
$$
According to this paper, equation 7.



It is assumed $t_0$ is the beginning of the time interval, at which $u(t)$ is at it's minimum and that $I(t)$ can vary. I have no idea where to even start, thank you.









share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 19 at 10:49









Rodrigo de Azevedo

12.6k41751




12.6k41751










asked Aug 19 at 10:45









John Miller

946




946











  • Could you please clarify what exactly are you asking? Do you want to understand how to solve LIF equation?
    – Evgeny
    Aug 20 at 9:52











  • Yes, or a step by step derivation of the general LIF so that I can understand it.
    – John Miller
    Aug 20 at 12:01
















  • Could you please clarify what exactly are you asking? Do you want to understand how to solve LIF equation?
    – Evgeny
    Aug 20 at 9:52











  • Yes, or a step by step derivation of the general LIF so that I can understand it.
    – John Miller
    Aug 20 at 12:01















Could you please clarify what exactly are you asking? Do you want to understand how to solve LIF equation?
– Evgeny
Aug 20 at 9:52





Could you please clarify what exactly are you asking? Do you want to understand how to solve LIF equation?
– Evgeny
Aug 20 at 9:52













Yes, or a step by step derivation of the general LIF so that I can understand it.
– John Miller
Aug 20 at 12:01




Yes, or a step by step derivation of the general LIF so that I can understand it.
– John Miller
Aug 20 at 12:01















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