Predictive Coding of Dynamical Variables in Balanced Spiking Networks
[['Neural systems need to integrate , store , and manipulate sensory information before acting upon it .', 'Various neurophysiological and psychophysical experiments have provided examples of how these feats are accomplished in the brain , from the integration of sensory stimuli to decision-making [1] , from the short-term storage of information [2] to the generation of movement sequences [3] .', 'At the same time , it has been far more difficult to pin down the precise mechanisms underlying these functions .', 'A lot of research on neural mechanisms has focused on studying neural networks in the framework of attractor dynamics [4]–[6] .', "These models generally assume that the system's state variables are represented by the instantaneous firing rates of neurons .", 'While quite successful in reproducing some features of electrophysiological data , these models have had a hard time reproducing the irregular , Poisson-like statistics of cortical spike trains .', 'A common assumption is that the random nature of spike times is averaged out over larger populations of neurons or longer periods of time [7]–[10] .', 'However , the biophysical sources of noise in individual neurons are insufficient to explain such variability [11]–[13] .', 'Several researchers have therefore suggested that irregular spike timing arises as a consequence of network dynamics [8] , [14] .', 'Indeed , large networks of leaky integrate-and-fire ( LIF ) neurons with balanced excitation and inhibition can be “chaotic” and generate asynchronous and Poisson-like firing statistics [15]–[18] .', 'While these studies explain how relatively deterministic single units can generate similar statistical properties as random spike generators in rate models , they generally do not clarify how particular computations can be carried out , nor do they fundamentally answer why the brain would be operating in such a regime .', 'Here we show that the properties of balanced networks can be derived from a single efficiency principle , which in turn allows us to design balanced networks that perform a wide variety of computations .', 'We start from the assumption that dynamical variables are encoded such that they can be extracted from output spike trains by simple synaptic integration .', "We then specify a loss function that measures the system's performance with respect to an idealized dynamical system .", 'We prescribe that neurons should only fire a spike if that decreases the loss function .', 'From these assumptions , we derive a recurrent network of LIF neurons that is able to implement any linear dynamical system .', 'We show that neurons in our network track a prediction error in their membrane potential and only fire a spike if that prediction error exceeds a certain value , a form of predictive coding .', 'Our work shows how the ideas of predictive coding with spikes , first laid out within a Bayesian framework [19] , 20 , can be generalized to design spiking neural networks that implement arbitrary linear dynamical systems .', 'Such multivariate dynamical systems are quite powerful and have remained a mainstay of control-engineering for real-world systems [21] .', 'Importantly , the networks maintain a tight balance between the excitatory and inhibitory currents received by each unit , as has been reported at several levels of cortical processing [22]–[26] .', 'The spike trains are asynchronous and irregular .', 'However , this variability is not noise: The neural population essentially acts as a deterministic “super-unit” , tracking the variable with quasi-perfect accuracy while each individual neuron appears to behave stochastically .', 'We illustrate our approach and its usefulness with several biologically relevant examples .'], ['Our basic model strategy is represented in Fig . 1 A . Let us consider a linear dynamical system describing the temporal evolution of a vector of dynamical variables , : ( 1 ) Here is the state transition matrix , and are time-varying , external inputs or command variables .', 'We want to build a neural network composed of neurons , taking initial state and commands as inputs , and reproducing the temporal trajectory of .', "Specifically , we want to be able to read an estimate of the dynamical variable from the network's spike trains .", 'These output spike trains are given by , where is the time of the spike in neuron .', "Our first assumption is that the estimate is obtained by a weighted , leaky integration of the spike trains , ( 2 ) where the matrix contains the decoding or output weights of all neurons , and is the read-out's decay rate .", 'Whenever neuron fires , a -function is added to its spike train , .', 'The integration of the respective delta-function contributes a decaying exponential kernel , , weighted by , to each dynamical variable , .', 'This contribution can be interpreted as a simplified postsynaptical potential ( PSP ) .', "The effect of a neuron's spike can be summarized by its weights , , which we call the output kernel of neuron .", 'Note that these weights correspond to the columns of the matrix .', "The estimate can also be written as a weighted linear summation of the neuron's firing rates , , if we define the time-varying firing rates of the neurons , , as ( 3 ) Our second assumption is that the network minimizes the distance between and by optimizing over the spike times , and not by changing the fixed output weight matrix .", 'This approach differs from the “liquid computing” approach in which recurrent networks have fixed , random connectivities while the decoding weights are learnt [27] .', 'In our case , the decoding weights are chosen a-priori .', 'In order to track the temporal evolution of as closely as possible , the network minimizes the cumulative mean-squared error between the variable and its estimate , while limiting the cost in spiking .', 'Thus , it minimizes the following cost function , ( 4 ) where denotes the Eucledian distance ( or L2 norm ) , and the Manhattan distance ( or L1 norm ) , which here is simply the sum over all firing rates , i . e . , .', 'Parameters and control the cost-accuracy tradeoff .', 'The linear cost term , controlled by , forces the network to perform the task with as few spikes as possible , whereas the quadratic cost term , controlled by , forces the network to distribute spikes more equally among neurons , as explained in Material and Methods .', 'To derive the network dynamics , we assume that the firing mechanism of the neurons performs a greedy minimization of the cost function .', 'More specifically , neuron fires a spike whenever this results in a decrease of , i . e . , whenever .', "As explained in Material and Methods , this prescription gives rise to the firing rule ( 5 ) with ( 6 ) ( 7 ) Since is a time-varying variable , whereas is a constant , we identify the former with the -th neuron's membrane potential , and the latter with its firing threshold .", 'In the limit , the membrane potential of the -th neuron can be understood as the projection of the prediction error onto the output kernel .', 'Whenever this projected prediction error exceeds a threshold , a new spike is fired , ensuring that precisely tracks .', 'For finite , the membrane voltage measures a penalized prediction error .', 'If the neuron is already firing at a high rate , only a correspondingly larger error will be able to exceed the threshold and lead to a spike .', 'To connect this firing rule with the desired network dynamics , Eqn .', "( 1 ) , we take the derivative of each neuron's membrane potential , Eqn .", '( 6 ) , and consider the limit of large networks ( see Material and Methods ) to obtain the differential equation ( 8 ) where is a leak term , is a weight matrix of connectivity filters , explained below , and corresponds to a white “background noise” with unit-variance .', 'The leak-term does not strictly follow from the derivation , but has been included for biological realism .', 'A similar rationale holds for the noise term which we add to capture unavoidable sources of stochasticity in biological neurons due to channel noise , background synaptic input , etc .', 'The differential equation then corresponds to a standard LIF neuron with leak term , external , feedforward synaptic inputs , recurrent synaptic inputs mediated through the weight matrix , and a firing threshold , as specified in Eqn .', '( 7 ) .', 'The weight matrix of connectivity filters is defined as ( 9 ) and contains both “fast” and “slow” lateral connections , given by the matrices ( 10 ) ( 11 ) where corresponds to the identity matrix .', 'Accordingly , the connectivity of the network is entirely derived from the output weight matrix , the desired dynamics , and the penalty parameter .', 'Note that the diagonal elements of implement a reset in membrane potential after each spike by .', 'With this self-reset , individual neurons become formally equivalent to LIF neurons .', 'Whereas the linear penalty , , influences only the thresholds of the LIF neurons , the quadratic penalty , , influences both the thresholds , resets , and dynamics of the individual neurons , through its impact on the diagonal elements of the connectivity matrix .', 'Slow and fast lateral connections have typically opposite effects on postsynaptic neurons , and thereby different roles to play .', 'The fast connections , or off-diagonal elements of the matrix , implement a competition among neurons with similar selectivity .', 'If neuron fires , the corresponding decreases in prediction errors ( ) are conveyed to all other neurons .', 'Neurons with similar kernels ( inhibit each other , while neurons with opposite kernels ( ) excite each other .', 'This is schematized by the blue and red connections in Fig . 1 A . In contrast , the slow connections , , implement a cooperation among neurons with similar selectivity .', 'These connections predict the future trajectory of ( term “” ) but also compensate for the loss of information due to the decoder leak ( term “” ) .', 'For example , when the variable is static ( , ) , these connections maintain persistent activity in the network , preventing the variable from decaying back to zero ( see below ) .', 'Note that when the internal dynamics of change on a slower time scale than the decoder ( i . e . ) , and if we neglect the cost term , slow and fast connections have the same profile , ( i . e . ) , but opposite signs .', 'The combined effect of fast and slow connections yields the effective PSPs in our network , , with , which can be obtained by integrating Eqn .', '( 8 ) for a single spike .', "Two example PSPs are shown in Fig . 1 B . We note that our network model may contain neurons that both inhibit and excite different targets , depending on the kernel sign , a violation of Dale's law .", 'This problem can be addressed by creating separate cost functions for excitatory and inhibitory neurons , as laid out in full detail in Text S1 .', 'Here , we simply interpret the resulting connectivity as the effective or functional connectivity of a network , akin to the types of connectivities arising in generalized linear models ( GLMs ) of neural networks [28] .', 'We now briefly consider how the above equations can be mapped onto realistic physical units .', 'This consideration has the additional benefit that it clarifies how the network parameters scale with the number of neurons ( see also Material and Methods ) .', 'In order to express the network dynamics in biophysically relevant units , the membrane potential , Eqn .', '( 6 ) , and threshold , Eqn .', '( 7 ) , have to be rescaled accordingly .', 'We can obtain proper membrane potential units in mV if we apply the simple transformations and .', 'In turn , we obtain the modified equations ( 12 ) ( 13 ) and the modified dynamics ( 14 ) with a resting potential of .', 'Note that both the feedforward and recurrent connectivities change in this case .', 'Specifically , we obtain and , and a similar expression for the noise , .', 'In turn , we can freely choose and to find realistic units .', 'For instance , we can fix the threshold at , and the reset potential at , which uniquely determines both and for each neuron .', 'In the absence of linear costs ( ) , the reset potential becomes simply .', 'When we increase the network size while keeping the average firing rates and the read-out constant , we need to change the decoding kernels .', 'Specifically , the decoding kernels need to scale with .', 'If we assume that the relative importance of the cost terms is held fixed for each neuron , then the original threshold scales with , and the original connectivities similarly scale with , compare Eqns .', '( 9–11 ) .', 'As a consequence , the rescaled synaptic weights , do not scale in size when the network becomes larger or smaller .', 'When considering the summation over the different input spike trains , we therefore see that all synaptic inputs into the network scale with : the feedforward inputs , the slow recurrent input , and the fast recurrent inputs ( the latter two are both contained in the matrix ) .', 'The equal scaling of all inputs maintains the detailed balance of excitation and inhibition in the network .', 'An instructive case is given if we neglect the cost terms for a moment ( ) in which case we obtain the following ( rescaled ) feedforward weights and connectivities: ( 15 ) ( 16 ) Accordingly , the strength of the lateral connections is independent of the kernel norm .', 'In contrast , the strength of the feed-forward connections scales with the inverse of the kernel norm .', 'Since smaller kernels provide a more precise representation , the precision of the rescaled network , and its firing rates , are controlled entirely by its input gain .', 'Once the dynamics and the decoders are chosen , Eqn .', '( 1 ) and Eqn .', '( 2 ) , the only free parameters of the model are , , , and .', 'The model presented previously can in principle implement any linear dynamical system .', 'We will first illustrate the approach with the simplest linear dynamical system possible , a leaky integration of noisy sensory inputs where can be interpreted as the sensory stimulus while represents shared sensory noise .', 'The corresponding dynamical system , Eqn .', '( 1 ) , is then given by ( 17 ) The integrated sensory signal is a scalar ( ) and represents the leak of the sensory integrator .', 'For a completely homogeneous network , in which the output kernels of all neurons are the same , we can solve the equations analytically which is shown in Text S1 .', 'A slightly more interesting case is shown in Fig . 1 C , D , which illustrate network dynamics for two different choices of .', 'Here we used neurons , half of them with positive kernels ( ) , and the other half with negative kernels ( ) .', 'Neurons with positive kernels fire when variable is positive or increases , while neurons with negative kernels fire when the variable is negative or decreases .', 'Moreover , we set the cost terms and at small values , ensuring that our objective function is dominated by the estimation error , compare Eqn .', '( 4 ) .', 'As a consequence , the estimate closely tracks the true variable .', 'Albeit small , the cost terms are crucial for generating biologically realistic spike trains .', 'Without them , a single neuron may for example fire at extremely high rates while all others stay completely silent .', 'The regularizing influence of the cost terms is described in more detail in Text S1 .', 'For , the network is a perfect integrator of a noisy sensory signal .', 'The neural activities resemble the firing rates of LIP neurons that integrate sensory information during a slow motion-discrimination task [1] .', 'In the absence of sensory stimulation , the network sustains a constant firing rate ( Fig . 1 C after sec ) , similar to line attractor networks [29]–[31] .', 'In fact , as long as the dynamics of the system are slower than the decoder ( ) , the instantaneous firing rates approximate a ( leaky ) integration of the sensory signals .', 'On the other hand , if the system varies faster than the decoder ( i . e . ) , then neural firing rates track the sensory signal , and model neurons have transient responses at the start or end of sensory stimulation , followed by a decay to a lower sustained rate ( Fig . 1 D ) .', 'These responses are similar to the adaptive and transient responses observed in most sensory areas .', 'The overall effect of the lateral connections depends on the relative time scales of the variable and the decoder ( Fig . 1 B ) .', 'For neurons with similar selectivity ( or equal read-out kernels , ) , the postsynaptic potentials are given by ( assuming ) , ( 18 ) For neurons with opposite read-out kernels , we obtain just a sign reversal .', 'When ( ) , the interplay of fast inhibition with slower excitation results in a bi-phasic interaction between neurons of similar selectivity ( Fig . 1 B , left ) .', 'Moreover , the network activity persists after the disappearance of the stimulus .', 'In the extreme case of the perfect integrator ( ) , the temporal integral of this PSP is exactly zero , which explains how the mean network activity can remain perfectly stable ( neither increase nor decrease ) in the absence of any sensory stimulation .', 'In contrast , lateral interactions are entirely inhibitory when the network tracks the stimulus on a faster time scale than the decoder ( i . e . , Fig . 1 B , right ) .', 'The dominance of lateral inhibition explains the transient nature of the network responses and the absence of persistent activity .', 'Other response properties of the model units are illustrated in Fig . 2 .', 'We define the tuning curves of the neurons as the mean spike count in response to a 1 s presentation of a constant stimulus .', 'Firing rates monotonically increase ( for positive kernels ) or decrease ( for negative kernels ) as a function of and are rectified at zero , resulting in rectified linear tuning curves ( Fig . 2 A ) .', 'Since all neurons have identical kernels ( i . e . all or ) , neurons with the same kernel signs have identical tuning curves .', 'However , such a homogeneous network is rather implausible since it assumes all-to-all lateral connectivity with identical weights , so that all units in the network receive exactly the same synaptic input and have the same membrane potential .', 'To move to more realistic and heterogeneous networks , we can choose randomized decoding kernels .', 'Even then , however , the connectivity matrix is strongly constrained .', 'For negligible costs , , the weight matrix has rank one ( since ) .', 'A lot more flexibility can be introduced in the network connections by simultaneously tracking variables with identical dynamics and identical control , rather than a single scalar variable .', 'Thus the variable and the kernels have dimensions and .', 'We then define the actual network output , , as the mean of those variables ( simply obtained by summing all network outputs ) .', 'The network estimation error , , is an upper bound on , ensuring similar performance as before ( see Fig . 3 ) .', 'Importantly , we can choose the output kernels to fit connectivity constraints imposed by biology .', 'For instance , the output kernels can be made random and sparse ( i . e . with many zero elements ) , resulting in random and sparse ( but symmetrical ) connection matrices .', 'In such a network , the tuning curves are still rectified-linear , but have different gains for different neurons ( Fig . 2 B ) .', 'Output spike trains of both homogeneous and inhomogeneous networks are asynchronous and highly variable from trial to trial ( see raster plots in Fig . 1 C , D and Fig . 2 ) .', 'Fano factors ( measured during periods of constant firing rates ) , CV1 , and CV2 , were all found to be narrowly distributed around one .', 'The interspike interval ( ISI ) distribution was close to exponential ( Fig . 2 C ) .', 'Moreover , noise correlations between neurons are extremely small and do not exceed 0 . 001 ( noise correlations are defined as the cross-correlation coefficient of spike count in a time window of 1 s in response to a constant variable ) .', 'Finally , analysis of auto and cross-correlograms reveals the presence of high-frequency oscillations at the level of the population ( Fig . 2 D ) .', 'These high frequency oscillations are not visible on Fig . 2 C since the size of the bin ( 5 ms ) is larger than the period of the oscillations ( 1 ms ) .', 'Note that if we add a realistic amount of jitter noise ( ms ) to spike times , these high frequency oscillations disappear without affecting the response properties of the network .', 'In contrast to the output spike trains , the membrane potentials of different neurons are highly correlated , since neurons with similar kernels ( ) receive highly correlated feed-forward and lateral inputs ( Fig . 4 A , B ) .', 'In the homogeneous networks without quadratic cost ( ) , these inputs are even identical , resulting in membrane potentials that only differ by the background noise ( Fig . 4 A ) .', 'Despite these strong correlations of the membrane potentials , the neurons fire rarely and asynchronously .', 'Fig . 4 C illustrates why this is the case: let us consider a population of neurons with identical output kernels , maintaining an estimate of a constant positive ( top panel , blue line ) .', 'Due to the decoder leak , the network needs to fire periodically in order to maintain its estimate at the level of ( top panel , red line ) .', 'However , the exact order at which the different neurons fire does not matter , since they all contribute equally .', 'The period between two spikes can be called an “integration cycle” .', 'Within one integration cycle , the prediction errors and thus the membrane potentials , , rise for all neurons ( bottom panel , red line ) .', 'Since all kernels are identical , however , all neurons compute the same prediction error and will reach their firing thresholds at approximately the same time .', 'Only chance ( in this case , the background noise ) will decide which neuron reaches threshold first .', 'This first neuron is the only one firing in this integration cycle ( middle panel , colored bars ) since it immediately inhibits itself and all other neurons .', 'This starts a new integration cycle .', 'As a result of this mechanism , while the population of neurons fire at regular intervals ( hence the high frequency oscillations in Fig . 2 D ) only one neuron fires in each cycle , and its identity is essentially random .', 'The resulting variability has no impact on the network estimate , since all spike orders give the same output .', 'In the presence of a quadratic cost ( ) , neurons that did not fire recently have a higher probability of reaching threshold first ( their membrane potential is not penalized by ) .', 'When the cost term is large compared to the background noise ( i . e . when , which is not the case in the example provided here ) , this tends to regularize the output spike trains and leads to s smaller than 1 .', 'However , this regularization is not observed in inhomogeneous networks .', 'The inhomogeneous network behaves similarly , except that all neurons do not receive the same inputs and do not reach threshold at the same time ( Fig . 4 B ) .', 'In this case , we can even dispense of the background noise ( i . e . ) since fluctuations due to past network activity will result in a different neuron reaching threshold first in each cycle .', 'The individual ups and downs caused by the synaptic inputs from other neurons will nonetheless appear like random noise when observing a single neuron ( Fig . 4 B , D ) .', 'Furthermore , even in this deterministic regime , the spike trains exhibit Poisson statistics .', 'In fact , changing the timing of a single spike results in a total reordering of later spikes , suggesting that the network is chaotic ( as illustrated in Fig . 3 ) .', 'We now apply this approach to the tracking of a 2D point-mass arm based on an efferent motor command .', 'The dynamical variable has dimensions corresponding to the arm positions and the arm velocities .', 'The arm dynamics are determined by elementary physics , so that ( 19 ) ( 20 ) where is a 2D ( control ) force exerted onto the arm , and captures possible friction forces .', 'To simulate this system , we studied an arm moving from a central position towards different equidistant targets .', 'This reaching out arm movement was obtained by “push-pull” control forces providing strong acceleration at the beginning of the movement , and fast deceleration at the end of the movement ( Fig . 5 A , top panel ) .', 'As previously , the network predicts the trajectory of the arm perfectly based on the forces exerted on it ( Fig . 5 A , bottom panel; we again use relatively small cost terms and ) .', 'The resulting spike trains are asynchronous , decorrelated , and Poisson-like , with unpredictable spike times ( rasters in Fig . 5 A; Fano factor and CVs close to 1 ) .', 'The membrane potential of neurons with similar kernels are correlated while output spike trains are asynchronous and decorrelated .', 'The effective postsynaptic potentials have biphasic shapes reflecting the integrative nature of the network for small friction forces ( ) .', 'To measure tuning curves in this “center out” reaching task , we varied the speed and direction of the movement , as well as the starting position of the arm .', 'Neural activity was defined as the mean spike count measured during movement .', 'As illustrated in Fig . 5 B , C , D , instantaneous firing rates are modulated by arm position , velocity and force .', 'We found that tuning curves to arm position are rectified linear , with varying thresholds and slopes ( as in Fig . 2 B ) .', 'Such linear-rectified gain curves with posture have been reported in premotor and motor cortical areas [32] , [33] .', 'In contrast , tuning curves to circular symmetric variables such as movement direction or arm angle are bell-shaped ( Fig . 5 B , C , D ) .', 'In addition , direction tuning curves are gain modulated by arm speed , such that responses are stronger for larger speed when the arm moves in the preferred direction , and weaker when the arm moves in the anti-preferred direction ( Fig . 5 B ) .', 'Finally , arm positions have both an additive and a gain modulating effect on the tuning curve , and these modulation can be monotonically increasing ( Fig . 5 C ) or decreasing ( Fig . 5 D ) with arm position .', 'These observations have a simple geometric explanation , schematized in Fig . 5 E for the velocity space , .', 'A neuron is maximally active ( ; assuming ) when its kernel ( , thick vector in Fig . 5 E ) points in the direction of the derivative of the prediction error , .', 'Since the decoder leak is faster than the arm dynamics , this error mostly points in the direction opposite to the leak , ( thin vectors ) .', "Within the velocity space , the kernel thus defines the neuron's preferred movement direction ( dashed line and filled circles ) .", "The neurons is less often recruited when the arm moves away from the kernel's direction ( empty circles ) , resulting in a bell-shaped tuning curve .", 'Finally , since the vector gets larger at larger speeds , more spikes are required to track the arm state resulting in a linear tuning to movement speed .', 'The same reasoning applies for the position space .', 'These predictions are independent of the choice of kernels and are in direct agreement with experimental data from the pre-motor and motor cortices [32] , [33] .', 'We chose to present a sensory integrator and an arm controller for their biological relevance and simplicity .', 'However , any linear dynamical system can be implemented within our framework , and our networks are not limited to performing integration .', 'To illustrate the generality of the approach , we applied the framework to two additional examples .', 'In Fig . 6 A , we simulated a “leaky differentiator” with a transition matrix .', 'This system of differential equations is designed so that the variable approximates a temporal derivative of a command signal .', 'The command signal , , is shown in the top panel of Fig . 6 A; the input signal is zero .', 'We used neurons with kernels drawn from a normal distribution , and then normalized to a constant norm of .', 'As in the other examples , the firing statistics are close to Poisson , with a .', 'In Fig . 6 B , we simulated a network that implements a damped harmonic oscillator .', 'Here we chose a transition matrix .', 'The oscillator is initially kicked out of its resting state through a force given by the command signal , as plotted on the top panel .', 'The input signal is zero .', 'We used neurons with kernels drawn from a normal distribution , and normalized to a constant norm of .', 'The network tracks the position and speed of the damped oscillator until position and speed are too close to zero to allow a reliable approximation .', 'The firing statistics of single units are again Poisson-like , with .', "Note that in these two examples , the dynamics implemented by the network are faster than the decoder's time scale .", 'Accordingly , our networks can track changes faster than the time scale of the decoder .', 'This speed-independence relies on a simple scheme: Spikes from neurons with positive kernel weight , , represent instantaneous increases in , whereas spikes from neurons with negative kernel weight represent instantaneous decreases in .', 'Even if the inter-spike interval is much shorter that , the decoder can therefore still provide an efficient staircase approximation for .', 'In conclusion , the temporal accuracy of these networks is not limited by , but by .'], ['Our current work both generalizes and modifies our earlier work in which we applied the principle of predictive coding with spikes to a Bayesian inference problem [20] .', 'This model tracked a log-probability distribution and implemented a non-linear drift-diffusion model , rather than a generic linear differential equation .', 'In addition , we here introduced cost terms which provided us with greater flexibility in regulating and controlling the dynamics of the spiking networks .', 'A quite general framework for designing networks of neurons that implement arbitrary dynamical systems has previously been described in the “neuro-engineering” approach [30] .', 'This approach relies on linearly combining the non-linear rate transfer function of LIF neurons .', 'In its essence , the method is therefore based on firing rates , and makes few predictions about the spiking statistics of cortical neurons .', 'A recently developed model , the “ReFiRe network” [34] provides a recipe for designing networks maintaining stable memories , and shares some of the features of our networks .', 'Just as the neuro-engineering framework , however , its design is essentially based on firing rates .', 'Here we have designed a network based on the principle of predictive coding with spikes .', 'Even though indistinguishable from older models on the single cell level , our work is different in several important respects .', 'A first major difference of our approach is that it predicts a detailed balance between excitation and inhibition , rather than imposing it upfront .', 'This balance follows from the attempt of the network to minimize the loss function , Eqn .', '( 4 ) , which in turn implies that the membrane potential of neurons represents a prediction error and that neurons spike only when this prediction error exceeds a certain value—a form of predictive coding .', 'Any increase in excitation causes an increase in prediction error , immediately compensated by an increase in inhibition to bring down the prediction error ( and vice versa ) .', 'This interplay causes a tight temporal correlation between excitation and inhibition at the time scale of the stimulus but also at a much finer time scale , within a single ISI ( Fig . 7 A ) .', 'Note that this balance only holds when considering all inputs .', 'In the leaky integrator , for instance , all lateral connections are inhibitory ( Fig . 1 B , right panel ) .', 'However , the network is still globally balanced when taking into account the contribution from the feedforward connections .', 'Such a tight balance between excitation and inhibition has been observed at several levels of cortical processing [22]–[26] .', 'Accordingly , spike trains in our network usually resemble independent Poisson processes , with rates tuned to the variable .', 'We note that spik
['Introduction', 'Results', 'Discussion', 'Materials and Methods']
['Two observations about the cortex have puzzled neuroscientists for a long time .', 'First , neural responses are highly variable .', 'Second , the level of excitation and inhibition received by each neuron is tightly balanced at all times .', 'Here , we demonstrate that both properties are necessary consequences of neural networks that represent information efficiently in their spikes .', 'We illustrate this insight with spiking networks that represent dynamical variables .', 'Our approach is based on two assumptions: We assume that information about dynamical variables can be read out linearly from neural spike trains , and we assume that neurons only fire a spike if that improves the representation of the dynamical variables .', 'Based on these assumptions , we derive a network of leaky integrate-and-fire neurons that is able to implement arbitrary linear dynamical systems .', 'We show that the membrane voltage of the neurons is equivalent to a prediction error about a common population-level signal .', 'Among other things , our approach allows us to construct an integrator network of spiking neurons that is robust against many perturbations .', 'Most importantly , neural variability in our networks cannot be equated to noise .', 'Despite exhibiting the same single unit properties as widely used population code models ( e . g . tuning curves , Poisson distributed spike trains ) , balanced networks are orders of magnitudes more reliable .', 'Our approach suggests that spikes do matter when considering how the brain computes , and that the reliability of cortical representations could have been strongly underestimated .']
['Two observations about the cortex have puzzled and fascinated neuroscientists for a long time .', 'First , neural responses are highly variable .', 'Second , the level of excitation and inhibition received by each neuron is tightly balanced at all times .', 'Here , we demonstrate that both properties are necessary consequences of neural networks representing information reliably and with a small number of spikes .', 'To achieve such efficiency , spikes of individual neurons must communicate prediction errors about a common population-level signal , automatically resulting in balanced excitation and inhibition and highly variable neural responses .', 'We illustrate our approach by focusing on the implementation of linear dynamical systems .', 'Among other things , this allows us to construct a network of spiking neurons that can integrate input signals , yet is robust against many perturbations .', 'Most importantly , our approach shows that neural variability cannot be equated to noise .', 'Despite exhibiting the same single unit properties as other widely used network models , our balanced networks are orders of magnitudes more reliable .', 'Our results suggest that the precision of cortical representations has been strongly underestimated .']
[]