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\title{Notes: Low-dimensional descriptions of the collective behavior of neural
populations}
\author[1]{Richard Gast}%
\affil[1]{Affiliation not available}%
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\date{\today}
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\section*{Montbrio et al. (2015) Macroscopic Description for Networks of
Spiking
Neurons}
{\label{885833}}\par\null
\subsection*{Summary}
{\label{976178}}
In this paper(
\url{https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.021028}),
Montbrio and colleagues derive a mathematically exact description of the
macroscopic dynamics of a globally coupled network of quadratic
integrate and fire neurons (QIFs). They start out with the membrane
potential evolution of a single QIF neuron:
\par\null\par\null
$$\tau \dot{v} = v^2 + I + \eta + Js \tau$$
where $\tau$ is the time-scale of the evolution, $I$ is an extrinsic current, $\eta$ is a background current that determines the exitability level, $J$ is a coupling constant and $s$ the synaptic activity.
Using the Lorentzian Ansantz (LA) and the Ott-Antonsen Ansatz (OA), they
derive the macroscopic descriptions for the average firing rate and
membrane potential of the population:
\par\null
$$\tau \dot{r} = \frac{\Delta}{\pi \tau} + 2 r v$$
$$\tau \dot{v} = v^2 + \overline{\eta} + I + J s \tau - (\tau \pi r)^2$$
where $\Delta$ and $\overline{\eta}$ are the parameters of a Lorenzian distribution over $\eta$ in the QIF population. They show that numerically and analytically that these equations are correct in the thermodynamic limit ($N \rightarrow \infty$).
\section*{Pietras \& Dattershofer (2016) Attractiveness of the
Ott-Antonsen Ansatz for Parameter-Dependent Oscillatory
Systems}
{\label{575768}}\par\null\par\null
In the thermodynamic limit ($N \rightarrow \infty$), the OA ansatz yields solutions for the dynamical evolution of the distribution of the oscillatory phases of a network of sinusiodally coupled oscillators that are attracted towards a reduced manifold of states. These oscillators can be described by:
$$\dot{\theta_j} = \omega_j + Im[He^{-i\theta_j}]$$
where $\theta$ is the phase of the oscillator, $\omega$ its natural frequency and $H$ a complex valued field that, if dependent on the mean field $z$, should express global coupling between all oscillators. Central to the OA ansatz is the description of the distribution function $\rho(\theta, \omega, t)$, where the quantity $rho(\theta, \omega, t)d\theta d\omega$ is the fraction of oscillators whose phases are in the range [\theta, \theta + d\theta] and have natural frequencies in [\omega, \omega + d\omega] at time t. The distribution function obeys the continuity equation:
$$\partial_t \rho + \partial_{\theta}(\rho \upsilon (\theta, t)) = 0$$
where $\upsilon(\theta, t)$ is the velocity field, equal to the right hand side of equation 1 when the subscript $j$ is dropped.
In this paper, Pietras and Dattershofer proof the low-dimensional attractiveness of the OA manifold for oscillator systems with intrinsic relationships between $\theta$, $H$ and $\omega$. This is expressed by the evolution equation for $\theta$:
$$\dot{\theta} = \Omega(\omega_j, \eta_j) + Im[H(\eta_j, t) e^{-i\theta_j}]$$
where $\eta_j$ is an additional parameter that the intrinsic frequency and the field $H$ may depend on. The latter can now also depend on time. Assuming that $\eta_j$ and $\omega_j$ are random, dependent variables drawn from nested probability distributions, this can be re-written as:
$$\partial_t \theta(\eta, t) = \omega (\eta) + Im[H(\eta, t) e^{-i\theta_j}]$$
Introducing a probability distribution $\rho(\theta, \eta, \omega, t)$ that satisfies the continuity condition, the velocity field $\upsilon$ is expressed by the above equation. After proving that the low-dimensional attractiveness of the OA manifold holds for such a system, they show that the Montbrio equations can be re-formulated to fit that scheme. Using $v_j = tan(\theta_j 0.5)$, they transform the QIF system into a theta neuron system:
$$\dot{\theta_j} = (1 - cos(\theta_j)) + (1 + cos(\theta_j))[\eta_j + J s(t) + I(t)]$$
which, considering the thermo-dynamic limit can be re-written as:
$$\partial_t \theta(\eta, t) = \omega (\eta) + Im[H(\eta, t) e^{-i\theta_j}]$$
with $H(\eta, t) = i[-1 + \eta + Js + I]$ and $\Omega(\eta, t) = \eta + Js + I + 1$.
Finally, they extend their own proof by loosening a couple of assumptions made earlier. I.e., they show that the dependence of $\omega$ on $\eta$ can be of any functional form $\omega = a(t)\eta + c(t)$ which can become non-linear for appropriate choices of $\dot{a}$ and $\dot[c}$. They show that their proof holds as long as $a(t)$ does not change signs. If that happens, the width of the distribution of $\omega$ (e.g. a Lorentzian) tends to zero and $\rho(\theta, \omega, t)$ exhibits a $\delta$ peak at this point in time. Furthermore, they show that the distribution of the additional parameter $\eta$ does not need to follow a Lorentzian distribution function, but could also be governed by a Gaussian for example. Last, but not least, they show that the global coupling condition can be loosened and that the attraction propoerties of the OA manifold also hold for heterogeneous couplings in the continuum limit $N \rightarrow \infty$.
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