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After 2 weeks of introduction, we finally move on to the energy trade-off in sensory adaptation. Sensory systems require adaptation processes to keep their sensory organs sensitive in different environments. This relies on negative feedback mechanisms and consumes energy especially in a noisy microscopic world. Does this set any limitations on our accuracy of adaptation? In this series, we will introduce the theoretical derivation of such limitation on a simple model system and do some computer simulations with the same model system.
Please make sure you have either read or understood the concepts of our previous episodes:
80 Fundamentals--Linear Stability Analysis
81 Entropy production described by Fokker-Planck equation
82 Computer simulation of entropy production
The materials of this series come from:
Lan G, Sartori P, Neumann S, Sourjik V, Tu Y (2012) The energy-speed-accuracy tradeoff in sensory adaptation. Nat Phys 8:422–428.
The derivation of detailed balance comes from:
https://physics.stackexchange.com/questions/153715/detailed-balance-condition-for-coupled-langevin-equation
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So now let's consider a simple sensory adaptation system composed of an input s, output a, and a negative feedback controller m.
+\eta_a(t)\\&space;\dot{m}=F_m(a,m)+\eta_m(t) "\\ \dot{a}=F_a(a,m,s)+\eta_a(t)\\ \dot{m}=F_m(a,m)+\eta_m(t)")
in which η represents our old friend -- white noise, which means
&space;\right&space;\rangle=0,\quad\left&space;\langle&space;\eta_a(t)\eta_a(t')&space;\right&space;\rangle=2\Delta_a\delta(t-t')\\&space;\left&space;\langle&space;\eta_m(t)&space;\right&space;\rangle=0,\quad\left&space;\langle&space;\eta_m(t)\eta_m(t')&space;\right&space;\rangle=2\Delta_m\delta(t-t') "\\ \left \langle \eta_a(t) \right \rangle=0,\quad\left \langle \eta_a(t)\eta_a(t') \right \rangle=2\Delta_a\delta(t-t')\\ \left \langle \eta_m(t) \right \rangle=0,\quad\left \langle \eta_m(t)\eta_m(t') \right \rangle=2\Delta_m\delta(t-t')")
Before even discussing the functionality of Fa & Fm, we will have to introduce the concept of detailed balance in this stochastic system. It really takes time to fully understand it so we will devote this episode fully to it.
What is detailed balance? For a system in equilibrium, there are many possible states: state a, state b, state c,... you name it. The system could change from state a to state b, or any other possibilities. However, for detailed balance to be true, the following must hold:
P(a\rightarrow&space;b)=\pi(b)P(b\rightarrow&space;a) "\pi(a)P(a\rightarrow b)=\pi(b)P(b\rightarrow a)")
Please make sure you have either read or understood the concepts of our previous episodes:
80 Fundamentals--Linear Stability Analysis
81 Entropy production described by Fokker-Planck equation
82 Computer simulation of entropy production
The materials of this series come from:
Lan G, Sartori P, Neumann S, Sourjik V, Tu Y (2012) The energy-speed-accuracy tradeoff in sensory adaptation. Nat Phys 8:422–428.
The derivation of detailed balance comes from:
https://physics.stackexchange.com/questions/153715/detailed-balance-condition-for-coupled-langevin-equation
/*-------------Divider-------------*/
So now let's consider a simple sensory adaptation system composed of an input s, output a, and a negative feedback controller m.
Figure source: Nat Phys 8:422–428. Fig. 1a. A schematic model of sensory adaptation system
The dynamics of this system could be modeled by a coupled Langevin equations:Before even discussing the functionality of Fa & Fm, we will have to introduce the concept of detailed balance in this stochastic system. It really takes time to fully understand it so we will devote this episode fully to it.
What is detailed balance? For a system in equilibrium, there are many possible states: state a, state b, state c,... you name it. The system could change from state a to state b, or any other possibilities. However, for detailed balance to be true, the following must hold:
The π means the probability density/mass function of each state, and P means the probability of the process that makes system change from state a to state b (this could be a collision, a chemical reaction, or whatever).
What is the detailed balance condition of our coupled Langevin equation? In our system, the state is defined by at least 2 variables -- a & m. The white noise terms are crazy and prevent us from writing down a simple equation. However, we could ask Fokker-Planck equation for some help since they are totally equivalent, and the Fokker-Planck equation is the governing equation of probability density of each state.
The Fokker-Planck equation of our system could be written as:
-\partial_m(F_mP)+\Delta_a\partial_a^2P+\Delta_m\partial_m^2P "\partial_tP=-\partial_a(F_aP)-\partial_m(F_mP)+\Delta_a\partial_a^2P+\Delta_m\partial_m^2P")
P(a',m')-W(a',m'|a,m)P(a,m)]\textup{d}a'\textup{d}m' "\partial_tP=\int\int[W(a,m|a',m')P(a',m')-W(a',m'|a,m)P(a,m)]\textup{d}a'\textup{d}m'")
What is the detailed balance condition of our coupled Langevin equation? In our system, the state is defined by at least 2 variables -- a & m. The white noise terms are crazy and prevent us from writing down a simple equation. However, we could ask Fokker-Planck equation for some help since they are totally equivalent, and the Fokker-Planck equation is the governing equation of probability density of each state.
The Fokker-Planck equation of our system could be written as:
However, from the fundamental meaning of probability flow, the Fokker-Planck equation could also be written as:
This is called the master equation.
This form is easy to interpret but looks a little bit ugly. We hope to express the master equation in this form:
P(a',m')\textup{d}a'\textup{d}m' "\partial_tP=\mathbb{W}P=\int\int\mathbb{W}(a,m|a',m')P(a',m')\textup{d}a'\textup{d}m'")
&space;-&space;\delta(a-a')\delta(m-m')\int\int&space;W(a'',m''|a,m)\textup{d}a''\textup{d}m'' "\mathbb{W} = W(a,m|a',m') - \delta(a-a')\delta(m-m')\int\int W(a'',m''|a,m)\textup{d}a''\textup{d}m''")
What is the detailed balance expressed in this form? The answer is obvious since detailed balance means the time evolution of probability density function vanishes:
P^{\textup{eq}}(a',m')=W(a',m'|a,m)P^{\textup{eq}}(a,m)\\&space;\partial_tP^{\textup{eq}}=\mathbb{W}P^{\textup{eq}}=0\\&space;\frac{W(a,m|a',m')}{P^{\textup{eq}}(a,m)}=\frac{W(a',m'|a,m)}{P^{\textup{eq}}(a',m')} "\\ W(a,m|a',m')P^{\textup{eq}}(a',m')=W(a',m'|a,m)P^{\textup{eq}}(a,m)\\ \partial_tP^{\textup{eq}}=\mathbb{W}P^{\textup{eq}}=0\\ \frac{W(a,m|a',m')}{P^{\textup{eq}}(a,m)}=\frac{W(a',m'|a,m)}{P^{\textup{eq}}(a',m')}")
}{P^{\textup{eq}}(a,m)}f(a,m)\textup{d}a\textup{d}m=\int\int\frac{W(a',m'|a,m)}{P^{\textup{eq}}(a',m')}f(a,m)\textup{d}a\textup{d}m "\int\int\frac{W(a,m|a',m')}{P^{\textup{eq}}(a,m)}f(a,m)\textup{d}a\textup{d}m=\int\int\frac{W(a',m'|a,m)}{P^{\textup{eq}}(a',m')}f(a,m)\textup{d}a\textup{d}m")
This form is easy to interpret but looks a little bit ugly. We hope to express the master equation in this form:
This requires the new operator to be:
What is the detailed balance expressed in this form? The answer is obvious since detailed balance means the time evolution of probability density function vanishes:
We discuss these detailed balance because we would like to know how these stuffs pose a limit, so we try to multiply these detailed balance condition with some arbitrary function f and them integrate it:
To preserve the symmetry of the equation, we multiply it with g(a',m') and integrate again.
}{P^{\textup{eq}}(a,m)}f(a,m)g(a',m')\textup{d}a\textup{d}m\textup{d}a'\textup{d}m'=\int\int\int\int\frac{W(a',m'|a,m)}{P^{\textup{eq}}(a',m')}f(a,m)g(a',m')\textup{d}a\textup{d}m\textup{d}a'\textup{d}m' "\int\int\int\int\frac{W(a,m|a',m')}{P^{\textup{eq}}(a,m)}f(a,m)g(a',m')\textup{d}a\textup{d}m\textup{d}a'\textup{d}m'=\int\int\int\int\frac{W(a',m'|a,m)}{P^{\textup{eq}}(a',m')}f(a,m)g(a',m')\textup{d}a\textup{d}m\textup{d}a'\textup{d}m'")
\iint\mathbb{W}(a,m|a',m')g(a',m')\textup{d}a'\textup{d}m'}{P^{\textup{eq}}(a,m)}\textup{d}a\textup{d}m=\iint\frac{g(a',m')\iint&space;\mathbb{W}(a',m'|a,m)f(a,m)\textup{d}a\textup{d}m}{P^{\textup{eq}}(a',m')}\textup{d}a'\textup{d}m' "\iint\frac{f(a,m)\iint\mathbb{W}(a,m|a',m')g(a',m')\textup{d}a'\textup{d}m'}{P^{\textup{eq}}(a,m)}\textup{d}a\textup{d}m=\iint\frac{g(a',m')\iint \mathbb{W}(a',m'|a,m)f(a,m)\textup{d}a\textup{d}m}{P^{\textup{eq}}(a',m')}\textup{d}a'\textup{d}m'")
After some complex rearrangement this will become
If we define inner product:
=\iint\frac{f(a,m)g(a,m)}{P^{\textup{eq}}(a,m)}\textup{d}a\textup{d}m "(f,g)=\iint\frac{f(a,m)g(a,m)}{P^{\textup{eq}}(a,m)}\textup{d}a\textup{d}m")
=(g,\mathbb{W}f) "(f,\mathbb{W}g)=(g,\mathbb{W}f)")
=\iint\frac{[-\partial_a(F_af)-\partial_m(F_mf)+\Delta_a\partial_a^2f+\Delta_m\partial_m^2f]g}{P^{\textup{eq}}}\textup{d}a\textup{d}m\\&space;=\iint[(F_af-\Delta_a\partial_af)\partial_a(\frac{g}{P^{\textup{eq}}})+(F_mf-\Delta_m\partial_mf)\partial_m(\frac{g}{P^{\textup{eq}}})]\textup{d}a\textup{d}m "\\(g,\mathbb{W}f)=\iint\frac{[-\partial_a(F_af)-\partial_m(F_mf)+\Delta_a\partial_a^2f+\Delta_m\partial_m^2f]g}{P^{\textup{eq}}}\textup{d}a\textup{d}m\\ =\iint[(F_af-\Delta_a\partial_af)\partial_a(\frac{g}{P^{\textup{eq}}})+(F_mf-\Delta_m\partial_mf)\partial_m(\frac{g}{P^{\textup{eq}}})]\textup{d}a\textup{d}m")
Then the above relation could be summarize as
Now back to our Fokker-Planck equation. By direct comparison, we would find what W means. So we could substitute it back to above relation: (with the help of integration by parts)
Do the same thing for (f, Wg), and we will get
\partial_a(\frac{g}{P^{\textup{eq}}})+(F_mf-\Delta_m\partial_mf)\partial_m(\frac{g}{P^{\textup{eq}}})=\\&space;(F_ag-\Delta_a\partial_ag)\partial_a(\frac{f}{P^{\textup{eq}}})+(F_mg-\Delta_m\partial_mg)\partial_m(\frac{f}{P^{\textup{eq}}}) "\\(F_af-\Delta_a\partial_af)\partial_a(\frac{g}{P^{\textup{eq}}})+(F_mf-\Delta_m\partial_mf)\partial_m(\frac{g}{P^{\textup{eq}}})=\\ (F_ag-\Delta_a\partial_ag)\partial_a(\frac{f}{P^{\textup{eq}}})+(F_mg-\Delta_m\partial_mg)\partial_m(\frac{f}{P^{\textup{eq}}})")
g(\partial_a\log&space;P^{\textup{eq}})+&space;F_mf\partial_mg+\Delta_m(\partial_mf)g(\partial_m\log&space;P^{\textup{eq}})&space;\\&space;=F_ag\partial_af+\Delta_a(\partial_ag)f(\partial_a\log&space;P^{\textup{eq}})+&space;F_mg\partial_mf+\Delta_m(\partial_mg)f(\partial_m\log&space;P^{\textup{eq}}) "\\F_af\partial_ag+\Delta_a(\partial_af)g(\partial_a\log P^{\textup{eq}})+ F_mf\partial_mg+\Delta_m(\partial_mf)g(\partial_m\log P^{\textup{eq}}) \\ =F_ag\partial_af+\Delta_a(\partial_ag)f(\partial_a\log P^{\textup{eq}})+ F_mg\partial_mf+\Delta_m(\partial_mg)f(\partial_m\log P^{\textup{eq}})")
Expand the above equation and do some math we will get
And finally we will get
(F_a-\Delta_a\partial_a\log&space;P^{\textup{eq}})+&space;(f\partial_mg-g\partial_mf)(F_m-\Delta_m\partial_m\log&space;P^{\textup{eq}})=0 "(f\partial_ag-g\partial_af)(F_a-\Delta_a\partial_a\log P^{\textup{eq}})+ (f\partial_mg-g\partial_mf)(F_m-\Delta_m\partial_m\log P^{\textup{eq}})=0")
Since both f & g are arbitrary functions, this requires
And that is the condition demanded by detailed balance. In our next episode, we will proceed to the discussion of the dynamic of the system. Stay tuned!
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