Locking of correlated neural activity to ongoing oscillations
To date it is unclear which channels the brain uses to represent and process information ., A rate-based view is argued for by the apparent stochasticity of firing 1 and by the high sensitivity of the network dynamics to single spikes 2 ., In an extreme view , correlated firing is a mere epiphenomenon of neurons being connected ., Indeed , a large body of literature has elucidated how correlations relate to the connectivity structure 3–14 ., But the matter is further complicated by the observation that firing rates and correlations tend to be co-modulated , as demonstrated experimentally and explained theoretically 4 , 5 ., If the brain employs correlated firing as a means to process or represent information , this requires in particular that the appearance of correlated events is modulated in a time-dependent manner ., Indeed , such modulations have been experimentally observed in relation to the expectation of the animal to receive task-relevant information 15 , 16 or in relation to attention 17 ., Oscillations are an extreme case of a time-dependent modulation of the firing rate of cells ., They are ubiquitously observed in diverse brain areas and typically involve the concerted activation of populations of neurons 18 ., They can therefore conveniently be studied in the local field potential ( LFP ) that represents a complementary window to the spiking activity of individual neurons or small groups thereof: It is composed of the superposition of the activity of hundreds of thousands to millions of neurons 19 , 20 and forward modeling studies have confirmed 21 that it is primarily driven by the synaptic inputs to the local network 22–24 ., As the LFP is a quantity that can be measured relatively easily , this mesoscopic signal is experimentally well documented ., Its interpretation is , however , still debated ., For example , changes in the amplitude of one of the components of the spectrum of the LFP have been attributed to changes in behavior ( cf . e . g . 25 ) ., A particular entanglement between rates and correlations is the correlated firing of spikes in pairs of neurons in relation to the phase of an ongoing oscillation ., With the above interpretation of the LFP primarily reflecting the input to the cells , it is not surprising that the mean firing rate of neurons may modulate in relation to this cycle ., The recurrent network model indeed confirms this expectation , as shown in Fig 1A ., It is , however , unclear if and by which mechanisms the covariance of firing follows the oscillatory cycle ., The simulation shown in Fig 1B indeed exhibits a modulation of the covariance between the activities of pairs of cells ., Such modulations have also been observed in experiments: Denker et al . 26 have shown that the synchronous activation of pairs of neurons within milliseconds preferentially appears at a certain phase of the oscillatory component of the LFP in the beta-range—in their words the spike-synchrony is “phase-locked” to the beta-range of the LFP ., They explain their data by a conceptual model , in which an increase in the local input , assumed to dominate the LFP , leads to the activation of cell assemblies ., The current work investigates an alternative hypothesis: We ask if a periodically-driven random network is sufficient to explain the time-dependent modulation of covariances between the activities of pairs of cells or whether additional structural features of the network are required to explain this experimental observation ., To investigate the mechanisms causing time-dependent covariances in an analytically tractable case , we here present the simplest model that we could come up with that captures the most important features: A local network receiving periodically changing external input ., The randomly connected neurons receive sinusoidally modulated input , interpreted as originating from other brain areas and mimicking the major source of the experimentally observed LFP ., While it is obvious that the mean activity in a network follows an imposed periodic stimulation , it is less so for covariances ., In the following we will address the question why they are modulated in time as well ., Extending the analysis of mean activities and covariances in the stationary state 13 , 27 , 28 , we here expose the fundamental mechanisms that shape covariances in periodically driven networks ., Our network model includes five fundamental properties of neuronal dynamics: First , we assume that the state of low and irregular activity in the network 1 is a consequence of its operation in the balanced state 29 , 30 , where negative feedback dynamically stabilizes the activity ., Second , we assume that each neuron receives a large number of synaptic inputs 31 , each of which only has a minor effect on the activation of the receiving cell , so that total synaptic input currents are close to Gaussian ., Third , we assume the neurons are activated in a threshold-like manner depending on their input ., Fourth , we assume a characteristic time scale τ that measures the duration of the influence a presynaptic neuron has on its postsynaptic targets ., Fifth , the output of the neuron is dichotomous or binary , spike or no spike , rather than continuous ., As a consequence , the variance of the single unit activity is a direct function of its mean ., We here show how each of the five above-mentioned fundamental properties of neuronal networks shape and give rise to the mechanisms that cause time-dependent covariances ., The presented analytical expressions for the linear response of covariances expose two different paths by which a time-dependence arises: By the modulation of single-unit variances and by the modulation of the linear gain resulting from the non-linearity of the neurons ., The interplay of negative recurrent feedback and direct external drive can cause resonant behavior of covariances even if mean activities are non-resonant ., Qualitatively , these results explain the modulation of synchrony in relation to oscillatory cycles that are observed in experiments , but a tight locking of synchronous events to a particular phase of the cycle is beyond the mechanisms found in the here-studied models ., To address our central question , whether a periodically-driven random network explains the experimental observations of time-modulated pairwise covariances , we consider a minimal model here ., It consists of one inhibitory ( I ) population and , in the latter part of the paper , additionally one excitatory population ( E ) of binary model neurons 6 , 27 , 29 , 32 ., Neurons within these populations are recurrently and randomly connected ., All neurons are driven by a global sinusoidal input mimicking the incoming oscillatory activity that is visible in the LFP , illustrated in Fig 2 . The local network may in addition receive input from an external excitatory population ( X ) , representing the surrounding of the local network ., The fluctuations imprinted by the external population , providing shared inputs to pairs of cells , in addition drive the pairwise covariances within the network 13 , c . f . especially the discussion ., Therefore we need the external population X to arrive at a realistic setting that includes all sources of covariances ., In the following , we extend the analysis of cumulants in networks of binary neurons presented in 6 , 13 , 27 , 28 , 33 to the time-dependent setting ., This formal analysis allows us to obtain analytical approximations for the experimentally observable quantities , such as pairwise covariances , that expose the mechanisms shaping correlated network activity ., Binary model neurons at each point in time are either inactive ni = 0 or active ni = 1 . The time evolution of the network follows the Glauber dynamics 34; the neurons are updated asynchronously ., At every infinitesimal time step dt , any neuron is chosen with probability d t τ ., After an update , neuron i is in the state 1 with the probability Fi ( n ) and in the 0-state with probability 1 − Fi ( n ) , where the activation function F is chosen to be, F i ( n ) = H h i - θ i h i = ∑ k = 1 N J i k n k + h extsinω t + ξ i H ( x ) = 1 if x ≥ 0 0 if x < 0 ., ( 1 ), We here introduced the connectivity matrix J with the synaptic weights J i j ∈ ℝ describing the influence of neuron j on neuron i ., The weight Jij is negative for an inhibitory neuron j and positive for an excitatory neuron ., Due to the synaptic coupling the outcome of the update of neuron i potentially depends on the state n = ( n1 , … , nN ) of all other neurons in the network ., Compared to the equations in 13 , page 4 , we added an external sinusoidal input to the neurons representing the influence of other cortical or subcortical areas and Gaussian uncorrelated noise with vanishing mean 〈ξi〉 = 0 and covariance 〈 ξ i ξ j 〉 = δ i j σ noise 2 . The threshold θi depends on the neuron type and will be chosen according to the desired mean activity ., We employ the neural simulation package NEST 35 , 36 for simulations ., Analytical results are obtained by mean-field theory 6 , 13 , 27 , 28 , 37 , 38 and are described for completeness and consistency of notation in the section “Methods” ., In the main text we only mention the main steps and assumptions entering the approximations ., The basic idea is to describe the time evolution of the Markov system in terms of its probability distribution p ( n , t ) ., Using the master Eq 14 we obtain ordinary differential equations ( ODEs ) for the moments of p ( n , t ) ., In particular we are interested in the population averaged mean activities mα , variances aα , and covariances cαβ, m α t ≔ 1 N α ∑ i ∈ α n i t ( 2 ), a α t ≔ 1 N α ∑ i ∈ α n i t - n i t 2 ( 3 ), c α β t ≔ 1 N α N β ∑ i ∈ α , j ∈ β , i ≠ j n i t n j t - n i t n j t , ( 4 ), which are defined as expectation values 〈〉 over realizations of the network activity , where the stochastic update of the neurons and the external noisy input presents the source of randomness in the network ., The dynamics couples moments of arbitrarily high order 33 ., To close this set of equations , we neglect cumulants of order higher than two , which also approximates the input by a Gaussian stochastic variable with cumulants that vanish for orders higher than two 39 ., This simplification can be justified by noticing that the number of neurons contributing to the input is large and their activity is weakly correlated , which makes the central limit theorem applicable ., In a homogeneous random network , on expectation there are Kαβ = pαβ Nβ synapses from population β to a neuron in population α ., Here pαβ is the connection probability; the probability that there is a synapse from any neuron in population β to a particular neuron in population α and Nα is the size of the population ., Mean Eq ( 2 ) and covariance Eq ( 4 ) then follow the coupled set of ordinary differential equations ( ODEs , see section II A in S1 Text for derivation ), τ d d t m α t = - m α t + φ ( μ α ( m t , h extsinω t ) , σ α ( m t , c t ) ) ( 5 ), τ d d t c α β t = { - c α β t + ∑ γ  S μ α m t , h extsinω t , σ α m t , c t × K α γ J α γ c γ β t + δ γ β a β t N β  } + α ↔ β , ( 6 ), where α ↔ β indicates the transposed term ., The Gaussian truncation employed here is parameterized by the mean μα and the variance σ α 2 of the summed input to a neuron in population α ., These , in turn , are functions of the mean activity and the covariance , given by Eqs ( 18 ) and ( 19 ) , respectively ., Here φ is the expectation value of the activation function , which is smooth , even though the activation function itself is a step function , therefore not even continuous ., The function φ fulfills limm → 0 φ = 0 and limm → 1 φ = 1 and monotonically increases ., Its derivative S with respect to μ has a single maximum and is largest for the mean input μ within a region with size σ around the threshold θ ., S measures the strength of the response to a slow input and is therefore termed susceptibility ., The definitions are given in “Methods” in Eqs ( 17 ) and ( 20 ) ., The stationary solution ( indicated by a bar ) of the ODEs Eqs ( 5 ) and ( 6 ) can be found by solving the equations, m ¯ = φ m ¯ ( 7 ), 2 c ¯ = S K J c ¯ + a ¯ N + transposed ( 8 ), numerically and self-consistently , as it was done in 13 , 27 , 33 ., The full time-dependent solution of Eqs ( 5 ) and ( 6 ) can , of course , be determined numerically without any further assumptions ., Besides the comparison with simulation results , this will give us a check for the subsequently applied linear perturbation theory ., The resulting analytical results allow the identification of the major mechanisms shaping the time-dependence of the first two cumulants ., To this end , we linearize the ODEs Eqs ( 5 ) and ( 6 ) around their stationary solutions ., We only keep the linear term of order hext of the deviation , justifying a Fourier ansatz for the solutions ., For the mean activities this results in m α ( t ) = m ¯ α + δ m α ( t ) = m ¯ α + M α 1 e i ω t with, M α 1 = ∑ β U α β M β 1 = ∑ β U α β h ext U - 1 S μ ¯ , σ ¯ β - i τ ω + 1 - λ β τ ω 2 + 1 - λ β 2 ., ( 9 ) The time-dependence of σ was neglected here , which can be justified for large networks ( “Methods” , Eqs ( 22 ) and ( 30 ) ) ., The matrix U represents the basis change that transforms W ¯ α β ≔ S ( μ ¯ α , σ ¯ α ) K α β J α β into a diagonal matrix with λα the corresponding eigenvalues ., We see that , independent of the number of populations or the detailed form of the connectivity matrix , the amplitude of the time-dependent part of the mean activities has the shape of a low-pass-filtered signal to first order in hext ., Therefore the phase of δm lags behind the external drive and its amplitude decreases asymptotically like 1 ω , as can be seen in Fig 3A and 3B ., If we also separate the covariances into their stationary part and a small deviation that is linear in the external drive , c α β ( t ) = c ¯ α β + δ c α β ( t ) , expand S ( μα ( t ) , σα ( t ) ) and a ( t ) around their stationary values , keeping only the terms of order hext and neglect contributions from the time-dependent variation of the variance of the input σ2 ( see “Methods” , especially Eq ( 30 ) for a discussion of this point ) , we get the ODE, τ d d t δ c t + 2 δ c t - W ¯ δ c t - W ¯ δ c t T = { W ¯ diag 1 - 2 m ¯ N diag δ m t ︸ modulated-autocorrelations-drive + diag K ⊛ J δ m t ︸ recurrent drive + h extsinω t ︸ direct drive diag ∂ S ∂ μ t K ⊛ J c ¯ total } + . . . T , ( 10 ), where we introduced the point-wise ( Hadamard ) product ⊛ of two matrices A and B see 40 , for a consistent notation of matrix operations as ( A ⊛ B ) ij ≔ AijBij , defined the matrix with the entries diag ( x ) ij := δij xi for the vector x = ( x1 , ‥ , xn ) and set c ¯ t o t a l ≔ c ¯ + d i a g ( a ¯ N ) to bring our main equation into a compact form ., We can now answer the question posed in the beginning: Why does a global periodic drive influence the cross covariances in the network at all and does not just make the mean activities oscillate ?, First , the variances are modulated with time , simply because they are determined via Eq ( 3 ) by the modulated mean activities ., A neuron i with modulated autocorrelation ai ( t ) projects via Jji to another neuron j and therefore shapes the pairwise correlation cji ( t ) in a time-dependent way ., We call this effect the “modulated-autocovariances-drive” , indicated by the curly brace in the second line of Eq ( 10 ) ., Its form in index notation is  W ¯ d i a g ( ( 1 − 2 m ¯ ) / N ) d i a g ( δ m ( t ) )  α β = W ¯ α β ( 1 − 2 m ¯ β ) / N β δ m β ( t ) ., This is the low-pass-filtered input ., The other contributions are a bit more subtle and less obvious , as they are absent in networks with a linear activation function ., The derivative of the expectation value of the activation function , the susceptibility , contributes linearly to the ODE of the covariances ., As the threshold-like activation function gives rise to a nonlinear dependence of φ on the mean input μ , the susceptibility S = φ′ is not constant , but depends on the instantaneous mean input ., The latter changes as a function of time by the direct external drive and by the recurrent feedback of the oscillating mean activity , indicated by the terms denoted by the curly braces in the third line of Eq ( 10 ) ., Together , we call these two term the “susceptibility terms” ., Both terms are of the same form,  diag δ μ ( t ) diag ∂ S ∂ μ t K ⊛ J c ¯ total  α β= δ μ α ( t ) ∂ S α ∂ μ α ∑ γ K α γ J α γ ( c ¯ γ β + δ γ β a ¯ β N β ) , ( 11 ), but with different δμα ., This form shows how the time-dependent modulation of the mean input δμα , by the second derivative of the gain function ∂ S α ∂ μ α = φ ″ , influences the transmission of covariances ., The sum following ∂ S α ∂ μ α is identical to the one in the static case Eq ( 8 ) ., For the “recurrent drive” , the time-dependent input is given by δμα ( t ) = ∑β Kαβ Jαβ δmβ ( t ) , which is a superposition of the time-dependent activities that project to population α and is therefore low-pass-filtered , too ., The term due to “direct drive” is δμα ( t ) = hext sin ( ωt ) ., We solve Eq ( 10 ) by transforming into the eigensystem of W ¯ and inserting a Fourier ansatz , δ c α β ( t ) = C α β 1 e i ω t ., The solution consists of a low-pass filtered part coming from the direct drive and two parts that are low-pass filtered twice , coming from the recurrent drive and the modulated-autocovariances-drive ., For a detailed derivation , consult the section “Covariances: Stationary part and response to a perturbation in linear order” ., We have calculated higher Fourier modes of the simulated network activity and of the numerical solution of the mean-field equations to check if they are small enough to be neglected , so that the response is dominated by the linear part ., Of course , it would be possible to derive analytical expressions for those as well ., However , we will see that the linear order and the corresponding first harmonic qualitatively and for remarkably large perturbations even quantitatively gives the right predictions ., The limits of this approximation are analyzed in Fig D in S1 Text ., We will therefore constrain our analysis to controlling the higher harmonics through the numerical solution ., In the following we will study three different models of balanced neuronal networks to expose the different mechanisms in their respective simplest setting ., We have left out so far several steps in the derivation of the results that were not necessary for the presentation of the main ideas ., In this section , we will therefore give a self-contained derivation of our results also necessitating paraphrases of some results known from earlier works ., The starting point is the master equation for the probability density of the possible network states emerging from the Glauber dynamics 34 described in “Binary network model and its mean field equations” ( see for the following also 13 , 37 ), ∂ p ∂ t ( n , t ) = 1 τ ︸ update rate ∑ i = 1 N ( 2 n i - 1 ) ︸ ∈ { - 1 , 1 } , direction of flux ϕ i ( n ∖ n i , t ) ︸ net flux due to neuron i ∀ n ∈ { 0 , 1 } N , ( 14 ), where, ϕ i ( n ∖ n i , t ) = p ( n i - , t ) F i ( n i - ) ︸ neuron i transition up - p ( n i + , t ) ( 1 - F i ( n i + ) ) ︸ neuron i transition down = - p ( n i + ) + p ( n i - , t ) F i ( n i - ) + p ( n i + , t ) F i ( n i + ) ., The activation function Fi ( n ) is given by Eq ( 1 ) ., Using the master equation ( for details cf . section II A in S1 Text ) , one can derive a differential equation for the mean activity of the neuron i , 〈ni〉 ( t ) = ∑n p ( n , t ) ni and the raw covariance of the neurons i and j , 〈ni ( t ) nj ( t ) 〉 = ∑n p ( n , t ) ninj 6 , 13 , 27 , 34 , 37 ., This yields, τ d d t n k t = - n k t + F k t d d t n k t n l t = - n k t n l t + n l t F k t + k ↔ l ., ( 15 ), As mentioned in “Binary network model and its mean field equations” , we assume that the input hi coming from the local and the external population is normally distributed , say with mean μi and standard deviation σi given by, μ i ( t ) ≔ h i = J n i + h ext sin ( ω t ) σ i 2 ( t ) ≔ h i 2 - h i 2 = ∑ k , k ′ = 1 N J i , k J i , k ′ n k n k ′ - n k n k ′ + σ i noise 2 = J T c J i i + J ⊛ J n ⊛ 1 - n + σ i noise 2 , ( 16 ), where the average 〈〉 is taken over realizations of the stochastic dynamics and we used the element-wise ( Hadamard ) product ( see main text ) ., The additional noise introduced in Eq ( 1 ) effectively leads to a smoothing of the neurons’ activation threshold and broadens the width of the input distribution ., It can be interpreted as additional variability coming from other brain areas ., Furthermore , it is computationally convenient , because the theory assumes the input to be a ( continuous ) Gaussian distribution , while in the simulation , the input ∑ l = k N J i k n k , being a sum of discrete binary variables , can only assume discrete values ., The smoothing by the additive noise therefore improves the agreement of the continuous theory with the discrete simulation ., Already weak external noise compared to the intrinsic noise is sufficient to obtain a quite smooth probability distribution of the input ( Fig 8 ) ., The description in terms of a coupled set of moment equations instead of the ODE for the full probability distribution here serves to reduce the dimensionality: It is sufficient to describe the time evolution of the moments on the population level , rather than on the level of individual units ., To this end we need to assume that the synaptic weights Jij only depend on the population α , β ∈ {exc ., , inh ., , ext ., } that i and j belong to , respectively , and thus ( re ) name them Jαβ ( homogeneity ) ., Furthermore , we assume that not all neurons are connected to each other , but that Kαβ is the number of incoming connections a neuron in population α receives from a neuron in population β ( fixed in-degree ) ., The incoming connections to each neuron are chosen randomly , uniformly distributed over all possible sending neurons ., This leads to expressions for the population averaged input hα , mean activity mα and covariance cαβ , formally nearly identical to those on the single cell level and analogous to those in 13 , sec . Mean-field solution ., The present work offers an extension of the well-known binary neuronal network model beyond the stationary case 6 , 13 , 27 , 28 , 33 ., We here describe the influence of a sinusoidally modulated input on the mean activities and the covariances to study the statistics of recurrently generated network activity in an oscillatory regime , ubiquitously observed in cortical activity 18 ., Comparing with the results of the simulation of the binary network with NEST 35 , 36 and the numerical solution of the full mean-field ODE , we are able to show that linear perturbation theory is sufficient to explain the most important effects occurring due to sinusoidal drive ., This enables us to understand the mechanisms by the help of analytical expressions and furthermore we can predict the network response to any time-dependent perturbation with existing Fourier representation by decomposing the perturbing input into its Fourier components ., We find that the amplitude of the modulation of the mean activity is of the order h ext / ( ( 1 − λ α ) 2 + ( τ ω ) 2 ) 1 2 , where λα , α ∈ {E , I} are the eigenvalues of the effective connectivity matrix W , i . e . the input is filtered by a first order low-pass filter and the amplitude of the modulation decays like ∝ ω−1 for large frequencies ., This finding is in line with earlier work on the network susceptibility 27 , esp . section V ., The qualitatively new result here is the identification of two distinct mechanisms by which the covariances δc are modulated in time ., First , covariances are driven by the direct modulation of the susceptibility S due to the time-dependent external input and by the recurrent input from the local network ., Second , time-modulated variances , analogous to their role in the stationary setting 13 , drive the pairwise covariances ., Our setup is the minimal network model , in which these effects can be observed—minimal in the sense that we would lose these properties if we further simplified the model: The presence of a nonlinearity in the neuronal dynamics , here assumed to be a threshold-like activation function , is required for the modulation of covariances by the time-dependent change of the effective gain ., In a linear rate model 10 , 46 this effect would be absent , because mean activities and covariances then become independent ., The second mechanism relies on the binary nature of neuronal signal transmission: the variance a ( t ) of the binary neuronal signal is , at each point in time , completely determined by its mean m ( t ) ., This very dependence provides the second mechanism by which the temporally modulated mean activity causes time-dependent covariances , because all fluctuations and therefore all covariances are driven by the variance a ( t ) ., Rate models have successfully been used to explain the smallness of pairwise covariances 6 by negative feedback 10 ., A crucial difference is that their state is continuous , rather than binary ., As a consequence , the above-mentioned fluctuations present due to the discrete nature of the neuronal signal transmission need to be added artificially: The pairwise statistics of spiking or binary networks are equivalent to the statistics of rate models with additive white noise 46 ., To obtain qualitative or even quantitative agreement of time-dependent covariances between spiking or binary networks and rate models , the variance of this additive noise needs to be chosen such that its variance is a function of the mean activity and its time derivative ., The direct modulation of the susceptibility S due to the time-dependent external input leads to a contribution to the covariances with first order low-pass filter characteristics that dominates the modulated covariances at large frequencies ., For small—and probably biologically realistic—frequencies ( typically the LFP shows oscillations in the β-range around 20 Hz ) , however , the modulation of the susceptibility by the local input from the network leads to an equally important additional modulation of the susceptibility ., The intrinsic fluctuations of the network activity are moreover driven by the time-dependent modulation of the variance , which is a function of the mean activity as well ., Because the mean activity follows the external drive in a low-pass filtered manner , the latter two contributions hence exhibit a second order low-pass-filter characteristics ., These contributions are therefore important at the small frequencies we are interested in here ., The two terms modulating the susceptibility , by the direct input and by the feedback of the mean activity through the network , have opposite signs in balanced networks ., In addition they have different frequency dependencies ., In networks in which the linearized connectivity has only real eigenvalues , these two properties together lead to their summed absolute value having a maximum ., Whether or not the total modulation of the covariance shows resonant behavior , however , depends also on the third term that stems from the modulated variances ., We find that in purely inhibitory networks , the resonance peak is typically overshadowed by the latter term ., This is because inhibitory feedback leads to negative average covariances 13 , which we show here reduce the driving force for the two resonant contributions ., In balanced E-I networks , the driving force is not reduced , so the resonant contribution can become dominant ., For the biologically motivated parameters used in the last setting studied here , the effective coupling matrix W has complex eigenvalues which cause resonant mean activities ., If the inhomogeneity was independent of the driving frequency , δc would have resonant modes with frequency fres and 2fres ., Due to the mixing of the different modes and by the frequency dependence of the inhomogeneity driving the modulation of covariances , these modes determine only the ballpark for the location of the resonance in the covariance ., Especially the resonances are not sharp enough so that each of them is visible in any combination of the modes ., Different behavior is expected near the critical point where ℜ ( λ ) ≲\u20091 . For predictions of experimental results , however , a more careful choice of reasonable biological parameters would be necessary ., In particular , the external drive should be gauged such that the modulations of the mean activities are in the experimentally observed range ., Still , our setup shows that the theory presented here works in the biologically plausible parameter range ., The goal of extracting fundamental mechanisms of time-dependent covariances guides the here presented choice of the level of detail of our model ., Earlier works 6 , 28 , 29 showed that our setup without sinusoidal drive is sufficient to qualitatively reproduce and explain phenomena observed in vivo , like high variability of neuronal activity and small covariances ., The latter point can be explained in binary networks by the suppression of fluctuations by inhibitory feedback , which is a general mechanism also applicable to other neuron models 10 and even finds application outside neuroscience , for example in electrical engineering 47 ., The high variability observed in binary networks can be explained by the network being in the balanced state , that robustly emerges in the presence of negative feedback 29 , 30 ., In this state , the mean excitatory and inhibitory synaptic inputs cancel so far that the summed input to a neuron fluctuates around its threshold ., This explanation holds also for other types of model networks and also for biological neural networks 48 ., We have seen here that the operation in the balanced state , at low frequencies , gives rise to a partial cancellation of the modulation of covariances ., Our assumption of a network of homogeneously connected binary neurons implements the general feature of neuronal networks that every neuron receives input from a macroscopic number of other neurons , letting the impact of a single synaptic afferent on the activation of a cell be small and the summed input be distributed close to Gaussian: For uncorrelated incoming activity , the ratio between the fluctuations caused by a single input and the fluctuations of the total input is N − 1 2 , independent of how synapses scale with N . However , the input to a neuron is actually not independent , but weakly correlated , with covariances decaying at least as fast as N−1 6 , 29 ., Therefore this additional contribution to the fluctuations also decays like N − 1 2 . The Gaussian approximation of the synaptic input relies crucially on these properties ., Dahmen et al . 39 investigated third order cumulants , the next order of non-Gaussian corrections to this approximation ., They found that the approximation has a small error even down to small networks of about 500 neurons and 50 synaptic inputs per neuron ., These estimates hold as long as all synaptic weights are of equal size ., For distributed synaptic amplitudes , in particular those following a wide or heavy-tailed distributions ( e . g . 49 , 50 , reviewed in 51 ) , we expect the simple mean-field approximation applied here to require corrections due to the strong effect of single synapses ., The generic feature of neuronal dynamics , the threshold-like nonlinearity that determines the activation of a neuron , is shared by the binary , the leaky integrate-and-fire and , approximately , also the Hodgkin-Huxley model neuron ., An important approximation entering our theory is the linearity of the dynamic response with respect to the perturbation ., We estimate the validity of our theory by comparison to direct simulations ., To estimate the breakdown of this approximation we compare the linear response to the first non-linear correcti
Population-wide oscillations are ubiquitously observed in mesoscopic signals of cortical activity ., In these network states a global oscillatory cycle modulates the propensity of neurons to fire ., Synchronous activation of neurons has been hypothesized to be a separate channel of signal processing information in the brain ., A salient question is therefore if and how oscillations interact with spike synchrony and in how far these channels can be considered separate ., Experiments indeed showed that correlated spiking co-modulates with the static firing rate and is also tightly locked to the phase of beta-oscillations ., While the dependence of correlations on the mean rate is well understood in feed-forward networks , it remains unclear why and by which mechanisms correlations tightly lock to an oscillatory cycle ., We here demonstrate that such correlated activation of pairs of neurons is qualitatively explained by periodically-driven random networks ., We identify the mechanisms by which covariances depend on a driving periodic stimulus ., Mean-field theory combined with linear response theory yields closed-form expressions for the cyclostationary mean activities and pairwise zero-time-lag covariances of binary recurrent random networks ., Two distinct mechanisms cause time-dependent covariances: the modulation of the susceptibility of single neurons ( via the external input and network feedback ) and the time-varying variances of single unit activities ., For some parameters , the effectively inhibitory recurrent feedback leads to resonant covariances even if mean activities show non-resonant behavior ., Our analytical results open the question of time-modulated synchronous activity to a quantitative analysis .
In network theory , statistics are often considered to be stationary ., While this assumption can be justified by experimental insights to some extent , it is often also made for reasons of simplicity ., However , the time-dependence of statistical measures do matter in many cases ., For example , time-dependent processes are examined for gene regulatory networks or networks of traders at stock markets ., Periodically changing activity of remote brain areas is visible in the local field potential ( LFP ) and its influence on the spiking activity is currently debated in neuroscience ., In experimental studies , however , it is often difficult to determine time-dependent statistics due to a lack of sufficient data representing the system at a certain time point ., Theoretical studies , in contrast , allow the assessment of the time dependent statistics with arbitrary precision ., We here extend the analysis of the correlation structure of a homogeneously connected EI-network consisting of binary model neurons to the case including a global sinusoidal input to the network ., We show that the time-dependence of the covariances—to first order—can be explained analytically ., We expose the mechanisms that modulate covariances in time and show how they are shaped by inhibitory recurrent network feedback and the low-pass characteristics of neurons ., These generic properties carry over to more realistic neuron models .
resonance frequency, perturbation theory, neural networks, random variables, neuroscience, covariance, probability distribution, mathematics, algebra, network analysis, quantum mechanics, computer and information sciences, animal cells, resonance, probability theory, physics, cellular neuroscience, cell biology, linear algebra, neurons, biology and life sciences, cellular types, physical sciences, eigenvalues