Deriving the static transfer function and the dynamical transfer function for the IF neuron

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Bridging spiking neuron models and rate models

Spiking neuron models offer insight into dynamics of local networks whereas neural rate models provide an insight into the global activity of networks. Both types of models exhibit qualitatively similar population dynamics such as multistability properties and the emergence of global oscillations. The rate models cannot be derived by averaging the slow variables in the network of spiking neurons [Ermentrout, 1994]. Recently, Ostojic and Brunel (2011) have shown that the decay in 1/f of the linear firing rate response of the EIF model makes it particularly suitable to perform an approximate reduction to a rate model. This is because the linear filter then resembles the Fourier transform of a decaying exponential function for high frequencies. To outline their derivation, let us consider a network of neurons that is stimulated with an input  current I(t) = I0 + (t) + Isn(t). Here, n(t) is a Gaussian process with zero mean, unit variance and correlation time s, Is is the standard deviation of the temporally correlated Gaussian noise, is the standard deviation of the temporally uncorrelated Gaussian noise, I0 is the static input current into the neurons and (t) is a white (no temporal correlations) Gaussian process with zero mean and unit variance i.e. h(t)(t0)i = (t − t0). The different realizations of the correlated Gaussian noise n(t) are generated through the equation: s dn(t) dt = −n(t) + (t).

Single population with oscillatory input

We have seen in the previous section (see section II.3) that the adaptive rate 0 model reproduces very well the firing rate dynamics of a single population of EIF neurons in the presence of a noisy input current I(t) = I0+(t)+Isn(t). It also reproduces well the autocorrelation of the population firing rates for a recurrent excitatory network (see figure II.5). Here, (t) is a white (no temporal correlations) Gaussian process with zero mean and unit variance, and n(t) is a Gaussian process with zero mean, unit variance and correlation time s. Note that the mean input current in both the cases was constant in time. In this section, will explore the limitations of this model by stimulating the recurrent excitatory network of EIF neurons with a mean input current that varies sinusoidally in time. We will consider the case of a network with zero coupling or finite excitatory coupling (as shown in figures II.6, II.7, II.8). We observe that the adaptive timescale rate model falls short of explaining the network activity in the presence of a varying sinusoidal current. We then define a new time scale to capture this behaviour (corresponding to figures II.9, II.10, II.11, II.12).
As explained before, for the network simulations, we use a network of Ne EIF neurons connected via delta synapses, whose membrane potential obeys the following equations.

Introducing a new timescale to rate model

To try and improve the estimate of the rate model as compared to the network model in the presence of a sinusoidal current of larger finite amplitudes, we introduce a new time constant to the rate model. The time constant is derived based on Augustin et al. [Augustin et al., 2016]. They derive the reduced LNexp model from the Fokker- Planck equation that describes the population activity of a network of sparsely coupled adaptive nonlinear integrate-and-fire neurons when exposed to a noisy synaptic input. They used a different method from the one described in section II.2, where the LN model used to approximate the Fokker-Planck equation has the linear filter which is the dynamic response of the firing rate. This linear filter D(t) was approximated with a single exponential of effective timescale eff which was derived from the amplitude decay at high frequencies of the Fourier transform of D(t). Instead, Augustin et al. derived the timescale by approximating D(t) with a fitted exponential (minimum least squared error between the fit and the original) for all frequencies. We compute the response rate function for the EIF neuron for all frequencies from 1 Hz to 1000 Hz with a linear spacing of 1Hz. This amounts to computing the Fourier transform of D(t). To fit it with an exponential, we approximate the Fourier transform with effA 1+i2feff for each value of input current I0 and compute the parameters of the fit which give the minimum squared error between the original data and the fit (see figure II.9). This gives a time constant that is a function of the input current as shown in blue in the figure II.10.

Dynamics of a single EI network

In this section, we first describe the dynamics of a network of two populations of neurons, excitatory (E) and inhibitory (I) neurons. We will look at the different dynamical regimes of this system and in the next section, we will make a connection to the two population rate model. We compare the dynamics of the EI network in different regimes with the corresponding adaptive rate model for the two populations.
We will then focus on the regime where this network displays oscillations. After we have identified this regime, we will spatially extend our EI network to two EI networks and study the different dynamical regimes in this extended model.
In the single EI network, the two populations i.e. the excitatory and inhibitory populations are connected to each other. In addition, the excitatory population is recurrently connected to itself (see figure III.1).

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Different dynamical regimes of the EI network

Using different values of the synaptic connectivity strengths, we can run the network simulations. If the network indeed converges to the firing rates that we have specified, then these firing rates are a stable point of the network for the corresponding connectivity strengths. If, on the other hand, the network does not converge to the stationary state of the specified firing rates: 5 Hz and 10 Hz for the excitatory population and the inhibitory population, respectively, in our case, then the fixed point is not stable for the corresponding parameters of the connectivity strengths. Indeed, in the network simulations, we find a range of parameters where the fixed point is stable (see figure III.2) and other points in the phase space of connectivity strengths where the network population firing rates either oscillate in time (see figure III.4) or converge to another fixed point (see figure III.3) or exhibit synchronous spikes of activity (see figure III.5).
Due to the Poissonian spiking of neurons, the population firing rate is not exactly 5 Hz but it fluctuates around it. To explain the different regimes on the phase diagram, we can turn to the corresponding adaptive rate models.

Table of contents :

Introduction
I Models of single neurons 
I.1 Hodgkin Huxley Model of a single neuron
I.2 Simplified single neuron models
I.2.1 Spike generation dynamics
I.2.2 Adaptation currents and spike rate adaptation
I.2.3 Refractoriness
I.2.4 Subthreshold voltage gated currents
I.2.5 Spatial structure of the cell
I.3 Synapses
I.3.1 Time course of synaptic currents
I.3.2 Synaptic plasticity
I.4 Dynamics of a single neuron instantaneous firing rate
I.4.1 Deriving the static transfer function and the dynamical transfer function for the IF neuron
I.5 Numerical implementation of the static transfer function and the dynamical transfer function
I.6 Analyzing network dynamics
I.6.1 Analyzing network dynamics using numerical simulations
I.6.2 Analyzing network dynamics using analytical calculations
I.6.3 Dynamics of Fully Connected Excitatory Networks
I.7 Conclusion
II Rate models – an introduction 
II.1 Rate Models
II.1.1 Single population models
II.1.2 Models with adaptation
II.1.3 Linear Nonlinear (LN) models
II.2 Bridging spiking neuron models and rate models
II.3 Autocorrelation of the firing rate
II.4 Single population with oscillatory input
II.5 Introducing a new timescale to rate model
II.6 Conclusion
IIIOscillations in EI networks 
III.1 Oscillations in the brain
III.2 Dynamics of a single EI network
III.2.1 Different dynamical regimes of the EI network
III.3 Rate Model with coupling
III.4 Comparing the oscillatory phases of the rate model and the network model
III.4.1 Computing the limit cycles of the EI dynamical system
III.4.2 Comparing the limit cycles of the two Adaptive rate models
III.5 Two coupled EI networks
III.5.1 Antiphase or finite phase different regime
III.5.2 Alternating phase regime
III.5.3 Modulating phase regime
III.5.4 Finite phase regime
III.5.5 Synchronous phase regime
III.5.6 Phase diagram for the two coupled EI groups
III.6 Analytical derivation of the bifurcation plot
III.6.1 Transition from the synchronous regime to the phase difference regime
III.6.2 Transition from finite phase difference regime to other regimes
III.6.3 Computing the phase difference regime at very low coupling factors
III.7 Effects of finite size noise in Adaptive rate model1
III.8 Comparing with the network simulations
III.9 Comparing with the finite size networks
III.10 Finite size networks with finite connectivity
III.11 Conclusion
IV Sensory Adaptation 
IV.1 Adaptation: Marr’s three levels of analysis
IV.1.1 Computational level
IV.1.2 Algorithmic level
IV.1.3 Implementation level
IV.2 Motion After Effect
IV.3 The zebrafish as a vertebrate model for systems neuroscience
IV.4 Optokinetic response and Motion After Effect
IV.5 Two-photon calcium imaging
IV.6 Visual system of zebrafish
IV.7 MAE in the zebrafish larva
IV.8 Laing Chow model
IV.9 MAE model in the zebrafish tectum
IV.9.1 Comparator cells
IV.10 Conclusion
V Sustained Rhythmic Brain Activity Underlies Visual Motion Perception in Zebrafish 
VI Conclusion and Future perspectives

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