An analytical model for the large, fluctuating synaptic conductance state typical of neocortical neurons in vivo

Transcription

1 An analytical model for the large, fluctuating synaptic conductance state typical of neocortical neurons in vivo Hamish Meffin Anthony N. Burkitt David B. Grayden The Bionic Ear Institute, Albert St East Melbourne, Vic 3002, Australia Correspondence: Tel , Fax Abstract A model of in vivo-like neocortical activity is studied analytically in relation to experimental data and other models in order to understand the essential mechanisms underlying such activity. The model consists of a network of sparsely connected excitatory and inhibitory integrate-and-fire (IF) neurons with conductance-based synapses. It is shown that the model produces values for five quantities characterizing in vivo activity that are in agreement with both experimental ranges and a computer-simulated Hodgkin-Huxley model adapted from the literature (Destexhe et al., 2001 Neurosci. 1067(1):13-24). The analytical model builds on a study by Brunel (J. Comput. Neurosci., : ), which used IF neurons with current-based synapses, and therefore does not account for the full range of experimental data. The present results suggest that the essential mechanism required to explain a range of data on in vivo neocortical activity is the conductance-based synapse and that the particular model of spike initiation used is not crucial. Thus the IF model with conductance-based synapses may provide a basis for the analytical study of the large, fluctuating synaptic conductance state typical of neocortical neurons in vivo.

2 Keywords: cortical background activity, conductance-based synapses, integrateand-fire neuron, analytical model.

3 1 Introduction Recordings from neocortical pyramidal cells show that the characteristics of these neurons in vivo are radically different to those in vitro. Foremost amongst these differences are the highly irregular spiking of these neurons and the many-fold increase in their conductance in vivo compared to in vitro. Studies of morphologically reconstructed, detailed biophysical models demonstrate that these differences can be explained by the impact of random cortical background activity upon the neuron, which causes the synaptic conductance to be large and to fluctuate rapidly. The model of Destexhe et al. (1999) has been particularly successful in this regard, as it has been able to reproduce the experimentally observed ranges of five quantities that characterize background activity, namely the mean spiking rate (ν = 1 to 20 Hz), the coefficient of variation of the inter-spike interval (CV 1), the mean (µ = -67 to -62 mv) and standard deviation (σ = 2 to 6 mv) of the membrane potential, and the effective membrane time constant (τ = 0.6 to 8 ms). Destexhe et al. (2001) have also developed a much simpler model that reduces the morphological complexity of the full model to a single point-conductance with Hodgkin-Huxley dynamics and have shown that this simpler model is also able to account for the in vivo data characterizing background activity. Further modelling studies indicate that this state of large, fluctuating synaptic conductance significantly alters the neurons integrative properties and may have a number of important functional implications for cortical processing including a role in modulating gain (Chance et al., 2002; Burkitt et al., 2003), allowing the rapid transmission of information through cortical areas (van Rossum et al., 2002; Panzeri et al., 2001) and being instrumental in the compensation of synaptic efficacy for dendritic attenuation (Rudolph & Destexhe, 2003b). Given the success of this semi-self-consistent account 1 in reproducing a variety of experimental data, even with simplified models, it is interesting to ask whether a 1 The account Destexhe et al. (2001) is semi-self-consistent in the sense that the input and output rates of the model neurons are both within the biologically plausible range, but the output rate does not determine the input rate. A truly self-consistent account, as developed in this paper, determines the statistics of the input from the statistics of the output, given the network structure. Here the mean and variance of the output and input inter-spike intervals are set to be equal. A more complex model would involve an account of synaptic depression and facilitation whereby the recent history of the synapse effects its efficacy.

4 model that is simple enough to permit analytical treatment is also capable of explaining the same range of data. Such a model could be useful for understanding the functional importance of this state of large, fluctuating synaptic conductances at the network level where computational complexity prohibits the use of biologically more detailed models in either simulation or analysis. It would also be useful for clarifying the mechanisms that are essential for maintenance of such a state. Some progress has been made towards developing such a model and associated analysis with binary neurons (van Vreeswijk & Somplinsky, 1996; van Vreeswijk & Somplinsky, 1998) and with integrate-and-fire (IF) neurons (Treves, 1993; Amit & Brunel, 1997; Brunel & Hakim, 1999; Brunel, 2000). The last of these studies appears to be the most biologically realistic to date, and was able to show through analytical means that a sparsely connected homogeneous network of excitatory and inhibitory model neurons had a stable stationary state with irregular spiking at rates consistent with cortical background activity, provided there was a modest excitatory external input and inhibition was stronger than excitation. However, the model used current-based rather than conductance-based synapses and thus was unable to account for the changes in conductance observed in vivo. It is demonstrated here that consequently the values predicted for τ, µ and σ lie outside the range observed experimentally for most parameter settings. If instead conductance-based synapses are introduced, it is shown, using the same analytical methods, that this discrepancy is rectified so that all five quantities characterizing background activity (ν, CV, τ, µ, σ) are within experimentally observed ranges over a wide domain of parameter settings, as is the case with biologically more detailed models. 2 Models Three neural models will be considered in this paper. The central model is the leaky integrate-and-fire neuron with conductance-based synapses. Analytical results will be obtained for this model and compared to results from two other models: the leaky integrate-and-fire neuron with current-based synapses (for which analytical results are available (Brunel, 2000)) and a point-conductance model with Hodgkin- Huxley dynamics (Destexhe et al., 2001) (for which simulation is necessary). Such a comparison allows the mechanisms that are essential for the maintenance of background activity to be identified. The basic equation describing the dynamics of the neurons membrane potential, V, for all three models can be summarized as c dv dt = I leak I spike I syn, (1)

5 where c is the membrane capacitance, I leak is the current due to the passive leak of the membrane, I spike is a current describing the spiking mechanism of the neurons, and I syn is a current describing the effect of synaptic input on the neuron. The leak current is the same for all three models: I leak = c (V V p) τ p, (2) with resting potential V p and passive membrane time constant τ p. The details of the other two currents are model-dependent and will be described below. The architecture of the network is the same for all models and consists of N neurons of which N e are excitatory and N i are inhibitory. Each neuron receives synapses from other neurons in the network, of which C e are excitatory and C i are inhibitory, as well as C x external excitatory synapses. The connections are sparse but numerous (1 [C e, C i ] N) and chosen at random. In this study we consider the case where excitatory and inhibitory neurons have identical characteristics 2. It is possible to treat the case where the properties are different 3, but, since this doubles the number of free parameters and the results obtained are not substantially different, these results are not presented here. 2.1 Integrate-and-fire neurons The two integrate-and-fire (IF) models considered in this paper share the same spiking mechanism, which can be formally described in terms of a current (see Eq. (1)), ( ) 1 dv I spike = c (V th V r )δ(v V th ). (3) dt V =V th Intuitively this states that when the membrane potential reaches threshold, V th, it fires and resets to V r (δ denotes the delta function). In addition, there is an absolute refractory period, τ r, before the membrane potential begins to evolve again according to Eq.(1). The two IF models have different synapses as discussed below. Conductance-based synapses. The IF neuron with conductance-based synapses is the primary focus of this study. Its synaptic current is described by I syn = ca k (V V k ) P k (t), (4) k=e,i,x and comprises contributions due to network activity from excitatory (P e (t)), inhibitory (P i (t)) and external excitatory (P x (t)) neurons. 2 corresponding to Model A in (Brunel, 2000). 3 Model B in (Brunel, 2000). The external input is

6 explicitly modelled as a temporally homogeneous Poisson process with constant intensity γ x = C x ν x, where ν x is the mean rate of a individual fibre. The recurrent inputs are determined by the firing times t m n of pre-synaptic neurons n, P k (t) = n m δ(t tm n D n ); k = e, i where D n is a transmission delay drawn from a distribution P d (D). V e and V i are the (constant) reversal potentials (V i V r < V th < V e ). The parameters a e, a i, and a x represent the integrated conductances over the time course of the synaptic event divided by the neural capacitance and are thus dimensionless. Current-based synapses. The synaptic current for a current-based synapse is independent of V (cf. Eq. (4)): I syn = k=e,i,x cb k P k (t). (5) b k is the change in potential as the result of a single synaptic event and cb k is the associated charge delivered to the neuron. (Technically, the firing times t m n in the processes P k (t) will be different for the two models, however this is not important in the following analysis and so for simplicity we use the same notation.) 2.2 The Hodgkin-Huxley neuron with conductance-based synapses The Hodgkin-Huxley (HH) model is adapted from Destexhe et al. (2001), which in turn was originally adapted from Destexhe & Paré (1999) and Traub & Miles (1991). In the interests of giving a clear and consolidated description, the full details are given in this document. The model comprises both a somatic and axonal compartment. Eq. (1) applies to each compartment, l = s, a (for soma and axon, respectively) and contains an additional axial current, I ax, connecting the two compartments c l dv l dt = I leak,l I spike,l I syn,l I ax,l (6) where I ax,a = (V a V s )/R ax = I ax,s, and R ax is the axial resistance. Both the spiking and synaptic currents differ from the IF models. For each compartment the spiking current is given by I spike = I Na + I Kd (7) I Na = g Na m 3 h(v V Na ), (8) I Kd = g Kd n 4 (V V Kd ), (9) where I Na and I Kd, are the Hodgkin-Huxley voltage dependent Na + current and delayed rectifier K + current, respectively, g Na and g Kd are the respective maximal

7 conductances for the Na + and K + conductances, and V Na and V Kd are the respective reversal potentials. m, h and n are the voltage dependent channel activation variables, whose dynamical equations are given in Appendix B. The synaptic current appears in the soma only, I syn,s I syn, and is similar to the conductance-based IF model except that finite synaptic time constants, τ k (k = e, i, x), are included. I syn = G k (t)(v V k ) (10) k=e,i,x dg k dt = G k τ k + g k P k (t) (11) where G k represents the lumped conductance of synapses of type k = e, i, x, and g k is the maximal change in conductance due to a single synaptic event. Adaptation currents, present in most excitatory cortical neurons and included in the Destexhe & Paré model (1999), have been omitted here for simplicity and to allow comparison with the IF models. At the low firing rates typical of cortical background activity their inclusion leads to results with minor quantitative differences but the same broad agreement with experimentally observed values over a wide range of parameters as for the HH model studied here. 2.3 Parameterization To facilitate the comparison between the three different models, it was important to carefully calibrate some parameters so that the models were as far as possible equivalent. There were three main aspects to this. First, consistency of the spiking mechanism across models was approximated on the basis of the Hodgkin-Huxley model for which the parameters given in Destexhe et al. (2001) were used (see Appendix A for values). Visual inspection of the intracellular trace (V (t)) for this model showed that for most stimuli the neuron had a threshold potential of around -55 mv and a reset potential of around -70 mv and consequently these values were used in the IF model for V th and V r, respectively 4. Other membrane potential parameters (V p, V i, and V e ) were set to be identical directly (see Appendix A). 4 This procedure is simple and gives approximate equivalence between models. Other procedures that attempt to take into account aspects such as the reset and threshold potential in both compartments or lingering after-spike K + conductances may give a better match between models, but were not pursued due to their complexity.

8 Second, consistency of the synaptic input across models was obtained by making the average charge delivered as the result of a single synaptic event approximately the same for all models. That is, q k = ca k ( V V k ) = cb k = g k τ k ( V V k ), (12) where V = (V th +V r )/2 is a fixed rough estimate of the average membrane potential 5, k = e, i, x. This equation was used to determine the values of a k, b k or g k, depending on the model. Finally, finite time constants were included in the Hodgkin-Huxley model because it was found that fluctuations in synaptic input resulting from the instantaneous synapses (delta-functions) used in the IF models are briefer than the duration of an action potential ( 1 ms) and thus interfere with the spiking mechanism, resulting in much reduced firing rates. Since real synapses have finite time constants, a small but finite value of τ k = 1 ms was used (k = e, i, x). While this value is on the low side of experimentally measured values, it does facilitate comparison with the conductance-based IF model presented here. The number of parameters is large so, following Brunel (2000), we fixed all but two of them, namely the ratio of inhibition and recurrent excitation, r = C iq i ν i C e q e ν e, (13) and the mean external spike-rate, ν x. Since C e, C i, and q e remain fixed in this study and both types of neurons are equivalent, so that ν e = ν i ν, Eq. (13) is here equivalent to varying the magnitude of q i. While some of the other parameters are significant, r and ν x are the most important and we chose to study their effect. The remaining parameters were fixed and take conventional or experimentally constrained values from the literature. Their values and source of reference are given in the Appendix A. 3 Analysis and methods 3.1 Integrate-and-fire neurons The analysis of the IF model with current-based synapses has been performed by Brunel (2000) and the analysis for the IF model with conductance-based synapses 5 A fixed estimate of the average membrane potential is required here rather than µ 0 since µ 0 varies with ν x and r (see Eq. (13)), which would imply that the synaptic conductance also varies in this way.

9 presented here proceeds along similar lines. The aim is to derive expressions for the five quantities (ν, CV, τ, µ, σ) that have been found experiments to characterize in vivo cortical activity of neurons. Under the assumptions that each neuron receives many small inputs per integration time (a k (V th V r )/( V V k ), k = e, i, x) and the network has sparse but numerous connectivity (1 [C e, C i ] N), the input P e (t) and P i (t) to a neuron may be replaced by Poisson processes with the same intensity (equal to γ e (t) = ν(t)c e and γ i (t) = ν(t)c i, respectively). These processes are in general not temporally homogeneous, and so may be correlated at the level of mean rates. However, the assumption of sparse connectivity implies that correlations between the firing times of different neurons are vanishingly small. Furthermore the Poision-like irregularity of neural firing during background activity implies that the correlations between firing times of the same neuron can be ignored, so that, in aggregate, the firing times may be regarded as independent subject to the prevailing rate. Finally, the additional assumption that the inputs are many and small allows the system to be described by the probability density of the membrane potential, P (V, t), which obeys a Fokker-Planck equation (van Kampen, 1992) P (V, t) τ(t) = σ 2 (t) 2 P (V, t) t V 2 + [(V µ(t))p (V, t)], (14) V where the effective membrane time constant and the mean and variance of the free membrane potential 6 are approximated 7 by (Burkitt, 2000) 1 = 1 + C k a k ν k (t) (15) τ(t) τ p k=e,i,x µ(t) = τ(t) V p + C k a k V k ν k (t) (16) τ p σ 2 (t) = τ(t) 2 k=e,i,x k=e,i,x C k a 2 k(v k µ(t)) 2 ν k (t), (17) where ν e (t) = ν i (t) = ν(t D)P d (D) dd and ν x (t) = ν x independent of time. The effective membrane time constant, τ(t), takes into account synaptic as well as passive contributions to the total membrane conductance (Bernander et al., 1991; Burkitt, 2000). It is a dynamic quantity that changes with the level of background activity impacting on synapses and thus it is a direct (though inverted) measure of 6 Free refers to the membrane potential that is not restricted by the threshold. 7 In Eq.(17) the V dependence of σ has been eliminated by replacing it with µ. Although it is possible to retain the full V dependence and still perform the analysis this does not improve the approximation, so the above approach is preferred as it gives formulae that are simpler and more intuitive.

10 the total synaptic conductance, k=e,i,x C ka k ν k (t), since the passive contribution does not change. The probability current is given by J(V, t) = σ2 (t) P (V, t) (V µ(t)) P (V, t) (18) τ(t) V τ(t) which leads to the following boundary conditions, P (V th, t) = 0 J(V th, t) = ν(t) P (V r +, t) P (Vr, t) = 0 J(V r +, t) J(Vr, t) = ν(t τ r ). The first condition is required to obtain a finite mean firing rate, the second so that the probability current through threshold is the mean firing rate, the third so that the distribution is continuous at reset, and the fourth so that the probability current across the reset is given by the mean firing rate subsequent to the refractory period. Last, the distribution should be integrable and satisfy the normalization condition, Vth t P (V, t)dv + ν(s)ds = 1. t τ r (19) The stationary solution, P 0 (V ), of Eq. (14) may be calculated, P 0 (V ) = 2 ν 0τ 0 exp ( (V µ 0) 2 ) yth Θ(u V r )e u2 du, (20) σ 0 2σ 2 0 where is Θ is the Heaviside function and y = (V µ 0 )/ 2 σ 0, y th = (V th µ 0 )/ 2 σ 0. The mean firing rate is obtained from the normalization condition, ( yth 1 ν 0 = τ r + τ 0 π e u 2 (1 + erf(u))du), (21) y r where erf is the error function and ν 0, τ 0, µ 0 and σ 0 are the stationary values of their respective variables. When supplemented with equations (15), (16), and (17), this is a non-linear self-consistent equation for ν 0. The coefficient of variation is given by (Tuckwell, 1988b), ( yth x ) 1 CV 0 = 2πν0τ e x 2 dx e y2 (1 + erf(y)) 2 2 dy. (22) y r The mean and the variance of the membrane potential, constrained by the threshold and excluding time spent in refractory, can be calculated 8 from Eq. (20), µ 0 = µ 0 τ 0ν 0 (V th V r ) 1 ν 0 τ r (23) 8 The time spent in refractory is excluded from the calculation by using a trick in which the mean and variance during this time are set to be identical to those during threshold-restricted evolution. y

11 σ0 2 = σ µ 2 0 µ 2 0 ν 0τ 0 (V th + V r + 2 µ 0 )(V th V r ). (24) 2(1 ν 0 τ r ) The preceding analysis gives formulae for five quantities characterizing in vivo-like neural activity (ν 0 Eq. (21), CV 0 Eq. (22), τ 0 Eq. (15), µ 0 Eq. (23), σ 0 Eq. (24)). Eqs. (20) to (24) have the same functional form as for the current-based IF neuron studied by Brunel (Brunel, 2000). The difference between the models appears through equations for τ, µ, and σ (Eq. (15) to (17)), which have a different functional dependence on ν. This difference manifests itself when performing a linear stability analysis of Eq. (14) by expanding about the stationary state with P (V, t) = P 0 (V ) + P 1 (V, t) and ν(t) = ν 0 + ν 1 (t) so that different equations are obtained (see Appendix C for details). In addition to determining the stability of the stationary solutions, the stability analysis also allows the Hopf-bifurcations to be identified, signifying a transition to oscillatory behavior. Stationary solutions, ν 0, of Eq. (21), the associated characteristics CV 0, τ 0, µ 0, and σ 0, and bifurcation curves 9 in the (r, ν x )-plane arising from Eqs.(21) and (49) (in Appendix C) were calculated numerically using the software package AUTO2000 (Doedel et al., 2000), which is specialized for this purpose. 3.2 Hodgkin-Huxley neurons with conductance-based synapses The stationary firing rate, ν 0, of the Hodgkin-Huxley model and the associated stationary characteristics CV 0, τ 0, µ 0, and σ 0 were calculated from simulations using a program (Destexhe, 2001) adapted from Destexhe et al. (2001) to run in the NEURON simulation environment (Hines & Carnevale, 1997). The only alterations made to the program were to eliminate an adaptation current, I M, to set the synaptic time constants to be both 1 ms instead of 2.7 ms and 10.5 ms for excitatory and inhibitory synapses, respectively, and to add code necessary to find the stationary value of the firing rate. The simulation makes use of the standard diffusion approximation whereby the Poisson processes in Eq. (11) are replaced by Gaussian white noise with the same mean and variance (Destexhe et al., 2001), dg k dt = G k E[G k ] τ k + 2Var[G k ] τ k X k (t). (25) 9 Bifurcation curves mark transitions in phase space where a solution either ceases to exist or looses stability to another solution. They delimit regions associated with qualitatively different types of behavior, however regions with qualitatively different types of behavior need not be separated by a bifurcation.

12 Here the mean and variance of the lumped conductance are given by E[G k ] = g k C k τ k νk in and Var[G k] = gk 2C kτ k νk in, respectively, and X k(t) is Gaussian white noise with zero mean and unit variance (k = e, i, x). The output spiking rate, ν out, given an input spiking rate, ν in = ν in e = νi in, was approximated from simulations by averaging over many (> 3000) output spikes. The stationary rate, ν 0 : ν out = ν in, was found using the bisection method (Press et al., 1992). CV 0, µ 0 and σ 0 were approximated from simulation by explicitly averaging over many output spikes with the stationary input rate, and τ 0 was calculated as 1 = 1 + c E[G k0 ], (26) τ 0 τ p k=e,i,x where E[G k0 ] are the stationary values of the mean lumped conductance of type k. 4 Results Results for the conductance-based IF, current-based IF, and the Hodgkin-Huxley models will be presented in turn, in each case comparing them to the experimental results from the literature and the preceding models. 4.1 Integrate-and-fire neurons with conductance-based synapses The analytical results from the IF neuron with conductance-based synapses are shown in column 1 of Fig.1 in relation to experimental data from the literature. Each of the five quantities characterizing background activity are plotted ((A) ν 0, (B) CV 0, (C) τ 0, (D) µ 0, and (E) σ 0 ) as a function of the ratio of inhibition to recurrent excitation, r, for four values of external input, ν x = {1, 2, 4, 8} ν th (where ν th = (V th V p )/(a x C x τ p ( V V x )) is the minimum external input required to bring the neuron to threshold with a constant current; ν th 1.4 Hz here). Also shown on each plot, in gray shade, are the experimentally observed ranges for the given quantity, with the limits given on the right ordinate. These ranges are the same as those used by Destexhe and Paré (1999) from their experimental study of spontaneous activity in cat neocortical pyramidal neurons in vivo (Paré et al., 1998). The exception is the value of τ 0, which we permit to be between 1 3 and 1 30 times the passive value instead of around 1 5, as given in (Destexhe & Paré, 1999), to accommodate other results showing a 1 20 to 1 30 decrease in conditions of high membrane resistance based on whole cell recordings (Pongracz et al., 1991; Spruston & Johnston, 1992). Generally it can be seen that there is a stable state of the network in which spiking

13 is at a low rate (ν = 1/τ r, Fig.1.A1), is irregular (CV 0 1, Fig.1.B1) and induces a large synaptic conductance (τ 0 τ p, Fig.1.C1, note log scale) as for cortical background activity. This occurs whenever inhibition is stronger than recurrent excitation (r > r c 1) and external input exceeds a minimal value required to sustain recurrent activity (ν x > ν x,c 0.8ν th ). The region of the (r, ν x )-plane that gives rise to this state has been marked on a bifurcation diagram in Fig.2.1 with the letter L (for low spiking rate, although other properties also characterize this state as described above). Also shown on this diagram are the regions that give rise to: quiescent behavior ( Q, ν x < ν x,c and r > r c ) for which ν 0 Hz; behavior in which the spiking rate is high and regular ( H ) (r < r c and ν x > ν x,c ); and bistable behavior involving both the H and the Q states. These states are incompatible with cortical background activity. The transition from the L state to the H is rapid but smooth, and thus is not mediated by a bifurcation. Column 1 of Fig.1 also shows that the analytical results agree with experimentally observed values for ν 0, CV 0, τ 0, µ 0, and σ 0 over a wide range of the two parameters r and ν x. The region of the (r, ν x )-plane for which all five quantities are within experimental ranges is shown in gray shade in Fig.2.1. It corresponds to a large subset of the region for which L state exists. It extends from r = 1, corresponding to exact balance between inhibition and recurrent excitation, to very large values of r ( 10, data not shown), and for ν x it is bound below by the experimental constraint that ν 0 > 1 Hz and above by the experimental constraint τ 0 > 0.6 ms. 4.2 Integrate-and-fire neurons with current-based synapses To highlight the importance of conductance-based synapses in the preceding results, we compare them to the results using current-based synapses in an IF neuron as previously derived by Brunel (2000). The five quantities characterizing background activity are plotted for the current-based IF neuron in column 2 of Fig.1 using the same format as for the conductance-based IF model already described. Overall the comparison reveals that while there is some similarity between the values of ν 0 and CV 0, the remaining three quantities τ 0, µ 0, and σ 0 have very different values. These similarities and differences are examined in detail below. The similarity of the current-based to the conductance-based model in the dependence of ν 0 and CV 0 on both r and ν x is exhibited in Fig.1.A2 (cf. A1) and Fig.1.B2 (cf. B1). These demonstrate that the same stable states, L, H and Q exist for the current-based IF model as were observed in the conductance-based coun-

14 terpart. There are some minor quantitative differences, most notably the slightly higher values of CV 0 for the current-based model, which exceed the experimental upper bound of 1.2 in the ν x = 8.0 ν th trace. The overall qualitative similarity is further emphasized in Fig.2.2 (cf. 2.1), which gives the bifurcation diagram for the current-based IF model. This shows that similar regions of the (r, ν x ) plane give rise to the various states L, H, Q and the bistable H,Q state. In particular the L state that is broadly consistent with cortical background activity occupies the same region in which inhibition is stronger than recurrent excitation (r > r c ) and external input exceeds the minimum value required to sustain recurrent activity (ν x > ν x,c ). In addition Fig.2.2 also shows a small thin region around the ν x = ν x,c boundary in which global oscillations in the mean firing rate occur (marked O ), while the stationary L is unstable in this region. This oscillatory state is extremely thin region of parameter space for the conductance-based IF model (0.82ν th < ν x < 0.83ν th ), bringing into question its significance. The oscillatory state arises in this region because the external input is only just enough to bring the neuron to threshold. Then, after a transmission delay, D (03) ms, the neural firing sends the network into a quiescent state (in this inhibitory-dominated regime), from which is recovers in a time primarily determined by the membrane time constant. The oscillatory region is comparatively small for the conductance-based IF neuron because the effective membrane time constant rapidly becomes of the same order as the transmission delay when the external input increases. The similarity between the current-based and conductance-based IF models does not extend to the values of τ 0, µ 0 and σ 0, as illustrated in Fig.1.C2, D2 and E2, respectively (cf. Fig.1.C1, D1 and E1). The most striking difference is that τ 0 is required to remain fixed at τ p independent of r or ν x in the current-based model (by definition), whereas it varies widely in the conductance-based model and is generally much smaller (note the log scale on the ordinate) due directly to the impact of activity on synaptic conductances. The reverse situation is true of µ 0 and σ 0, which vary widely for the current-based IF model, but very little for the conductance-based IF model as r and ν x change. These observations demonstrate that the similarity in ν 0 and CV 0, noted above, are mediated by different mechanisms. Eq. (21) and (22) indicate that these two quantities are functions of τ 0, µ 0, and σ 0 only. Thus in the case of the conductance-based IF model, changes in ν 0 and CV 0 are brought about primarily by changes in τ 0, while in the case of the current-based IF model these changes are primarily brought about by the joint variation of µ 0 and σ 0. The result is that the current-based model is does not conform to the experimentally observed

15 ranges for most values of r and ν x. This is indicated by absence of any gray shaded region in Fig.2.2 that would correspond to values of (r, ν x ) that are consistent with experimental observations for all five quantities ν 0, CV 0, τ 0, µ 0, and σ Hodgkin-Huxley neurons with conductance-based synapses The results of simulations of the conductance-based HH neuron to determine the self-consistent value of ν 0 and the associated values of CV 0, τ 0, µ 0, and σ 0 are shown in column 3 of Fig.1 in the same format as described previously for the other two models. All five quantities show broad agreement with both the experimentally observed ranges and the conductance-based IF model in the region of the (r, ν x ) plane associated with cortical background activity (previously termed the L state). However, there are some quantitative differences and the behavior of the HH neuron in the regime in which recurrent excitation is stronger than inhibition is entirely different to that of the IF models. Fig.1.A3, B3 and C3 indicate that the conductance-based HH neuron supports a state with low rate, irregular spiking and large synaptic conductance whenever inhibition is stronger than recurrent excitation and the external input exceeds the minimum value required to sustain recurrent activity. The qualitative trends of ν 0, CV 0, τ 0, µ 0 and σ 0 (Fig.1.A3, B3, C3, C3 and E3, respectively) are similar to the conductancebased IF model in this region. The quantitative values are also similar in the sense that for a large part of this region, the values of all five quantities are within experimental bounds, as illustrated by the gray shaded region in the (r, ν x )-plane of Fig.2.3. This region has a similar extent to the conductance-based IF model, extending from r 1 to large values of r, and for ν x being bounded below by the experimental constraint ν 0 > 1.0 Hz. The upper bound is in a roughly similar position to that of the conductance-based IF model, but is formed by the µ 0 < 62 mv experimental constraint, rather than the constraint of τ 0 > 0.6 in the case of the IF model. The similarity of the HH and conductance-based IF models does not extended into the region for which recurrent excitation dominates inhibition (r < 1). There is a much more gradual transition of ν 0 towards higher values, and CV 0 and τ 0 toward lower values in the vicinity of r = 1 than was the case for the IF models. Indeed the mean firing rate remains below 100 Hz for r as low as around 0.5, at which point a transition occurs so that for r less than this, the membrane potential oscillates rapidly about -45 mv without producing an action potential. This region is marked

16 N (for non-firing) in Fig.2.3. However for larger values of r, roughly between 0.5 and 0.75, action potentials are discernable but with much reduced amplitude. The mechanism for the non-firing behavior has not been investigated but is likely to involve the chronic activation of the Na + and K + voltage-dependent conductances responsible for spike generation. A final point is that the stability of the stationary states in the HH model remain undetermined because this would require either the simulation of a very large network or an analytical treatment, both of which appear infeasible presently. The similarity to the conductance-based IF model in the inhibition-dominated regime, for which stability has been demonstrated analytically, indicates that in this regime the HH model may be stable too. 5 Discussion & Conclusions The results presented here show that a network of sparsely connected IF neurons with conductance-based synapses has a regime with low-rate (ν 0 τr 1 ), irregular (CV 0 1) spiking activity and large synaptic conductances (τ 0 τ p ) that is consistent with experimental data on five quantities (ν 0, CV 0, τ 0, µ 0, σ 0 ) characterizing in vivo-like activity (Fig.1,A1-E1). It is able to do so over a wide range of parameters, corresponding to situations in which inhibition is stronger than recurrent excitation and excitatory external input is neither too weak nor too strong (see Fig.2.1). These results are similar to another model with conductance-based synapses, also considered here, which incorporates a Hodgkin-Huxley rather than integrate-and-fire spiking mechanism. This biologically more sophisticated model is also able to reproduce the values for the five characteristic quantities, as originally demonstrated by Destexhe et al. (2001), and over a similar range of parameters (see Fig.2.3). In contrast a model that uses current-based synapses (and an IF spiking mechanism (Brunel, 2000)) has no mechanism by which to change conductance and thus cannot account for all the experimental data on these quantities for any parameter values (see Fig.2.2). Taken together, these results indicate that the essential mechanism required to explain the basic data characterizing in vivo-like neo-cortical activity is the conductance-based synapse (as previously emphasized by Tiesinga et al. (2000) and Salinas & Sejnowski (2000) in the context of a single neuron), and that the particular model of spike initiation is not crucial. This point is further emphasized when one considers the many things that are omit-

17 ted from all the models considered here, including finite synaptic time constants, non-sparse connectivity, non-homogeneous and structured synaptic efficacies, correlations in the times of spikes, adaptation of neural firing, synaptic adaptation and failure, dendritic morphology and so on. These factors are all likely to have some effect on ν 0, CV 0, τ 0, µ 0 and σ 0. Some of these factors (correlations, firing adaptation and dendritic morphology) have been included in the detailed biophysical model of Destexhe & Paré (1999), which produced results that differed in detail, but not in essence, from those described here for the models with conductance-based synapses. This suggests that the inclusion of further factors may result in marginal changes to the region of parameter space that is consistent with the experimental ranges for ν 0, CV 0, τ 0, µ 0 and σ 0, but will not alter the essential consistency of the model with these data. In this case, the present IF model with conductance-based synapses is a minimal model required to capture the essential aspects of these data on in vivo-like activity. The advantage of this model is its simplicity and consequent analytical tractability. 5.1 Implications for the mechanisms underlying irregular firing during cortical background activity Many studies have attempted to use the irregular firing of cortical neurons in vivo to make inferences about the cortical code and cortical processing. Softky & Koch (1993) pointed out that the integration of many randomly arriving excitatory postsynaptic potentials leads to highly regular, rather than irregular, neural firing. Given evidence that the spiking mechanism is very reliable, they suggested that either the synaptic input must be correlated, leading to irregular fluctuations in the membrane potential, or there must be some strong dendritic non-linearities in neural processing in order to achieve irregular firing, especially at high rates ( Hz). Subsequently Shadlen & Newsome (1994; 1998) showed that highly irregular firing also results from either a short membrane time constant, so that the neuron behaves as a coincidence detector responding to random variations in the membrane potential, or from a balance between excitation and inhibition, which leads to a subthreshold mean membrane potential in which small fluctuations sporadically bring the neuron over threshold. They argued that since the (passive) membrane time constant is too long to produce the required irregularity, a balance between excitation and inhibition is the likely cause of the irregular firing. Other suggestions for obtaining a sufficient degree of firing irregularity have included a partial reset of the membrane potential after firing (Troyer & Miller, 1997; Bugmann et al.,

18 1997) and the interaction of random synaptic input with non-linearities in type I Hodgkin-Huxley spike dynamics (Gutkin & Ermentrout, 1998). Some of these possible mechanisms are discussed below in relation to the present results. It should be noted that the result presented here pertain to low rates (1-20 Hz) typical of cortical background activity. The extent to which the same mechanisms may apply to irregularity at the high rates ( Hz) consider by Softky and Koch (1993) is not addressed here. A number of studies have reported that some degree of correlation in the inputs results in values of CV that are consistent with experimental data, whereas an absence of input correlations gives values that are too low (Softky & Koch, 1993; Destexhe & Paré, 1999; Destexhe et al., 2001; Salinas & Sejnowski, 2000; Stevens & Zador, 1998). While this finding is not supported by the present study (correlations were not present and not necessary), this may be because of the low rates considered, or the very short synaptic time constants used (instantaneous in the IF models and 1 ms in the HH model), which have a similar effect to introducing weak correlations together with more realistically long time constants. Certainly there is mounting evidence from a variety of sources that there are significant correlations between the firing times of cortical neurons in vivo (Arieli et al., 1996; Tsodyks et al., 1999; Azouz & Gray, 1999; Lampl et al., 1999; Usher et al., 1994). The effect of such correlations was not considered in this study, but it would of interest to incorporate them into the analysis in a self-consistent manner in order to address the data on this issue. Some progress has already been made in providing a first order correction 10 to the uncorrelated case for an isolated IF neuron with currentbased synapses (Brunel & Sergi, 1998; Moreno et al., 2002). A full analysis of the problem will require two steps: first, extending these results to conductance-based synapses with higher order corrections, and second, the development of methods for self-consistently establishing, both mean and correlated activity of a network of neurons with a specified architecture (this problem was avoided in the present study by assuming sparse connectivity; see (Usher et al., 1994; Tsodyks & Sejnowski, 1995) for simulation studies addressing this problem). There have also been many recent studies that demonstrate the existence of voltagegated conductances in the dendrites of cortical neurons (see (Johnston et al., 1996; Stuart et al., 1999; Häusser et al., 2000) for reviews). The non-linear effects on 10 the expansion parameter was the square root of the ratio of the synaptic to the membrane time constants

19 neuronal integration caused by such conductances under in vivo conditions are only beginning to be explored with the aid of multi-compartmental biophysical models (Rudolph & Destexhe, 2003b; Rudolph & Destexhe, 2003c) and experimental techniques (see (Stuart et al., 1999) for an overview). Some studies have indicated the effect of these conductances in cortical pyramidal neurons may be to counter attenuation of PSPs due to their (distal) dendritic location (Williams & Stuart, 2000; Williams & Stuart, 2002; Rudolph & Destexhe, 2003b). In this case the IF point-neuron may provided an adequate model to capture the essence of cortical activity. However, there are also indications that significant non-linear interactions may occur within and between various compartments of a neuron (Mel, 1999), such as result from dendritically initiated spikes. In this case, although the neuronal models studied here (with conductance-based synapses) can account for basic in vivo data on ν 0, CV 0, τ 0, µ 0 and σ 0, it is unlikely that they will be adequate for the study of many aspects of cortical network activity and function. Other simplified models will need to be developed to permit the computationally feasible study of large networks of cortical neurons. The issue of balance between excitation and inhibition in cortical neurons has also received a lot of attention recently both as a mechanism to explain irregular spiking activity and for its possible functional implications. Following Shadlen and Newsome (Shadlen & Newsome, 1994; Shadlen & Newsome, 1998) and pioneering work (Gerstein & Mandelbrot, 1964; Tuckwell, 1988a), reports using isolated IF neurons with current-based synapses found the need for an approximate balance between excitation and inhibition to achieve sufficient irregularity in the firing of model neurons (Bugmann et al., 1997). Later reports using conductance-based synapses in either IF neurons, HH point-neurons or multi-compartmental biophysical models indicated that the required degree of irregularity could be achieved over a relatively wide range of ratios of inhibition to excitation. The results presented here address this question from a different perspective. Here the ratio of inhibition to recurrent excitation is the independent parameter, while the ratio of inhibition to total excitation, r tot, is determined by the behavior of the network as a whole through feedback and is not a free parameter. As shown Fig.1 row B, CV 0 1 for all three models provided the ratio of inhibition to recurrent excitation is greater than 1. However when r tot is considered, there is a greater distinction between the models, in agreement with the previous studies mentioned above. This is demonstrated in Fig.3 row A, which shows r tot as a function of r for each of the three different models. For the IF model with current-based synapses (Fig.3.A2), r tot is more tightly constrained

20 (by the network) to values around 1 than either of the conductance-based models, indicating a greater degree of balance. In contrast, the balance between inhibition and excitation can change considerably for the models with conductance-based synapses, depending of the level of external excitatory drive, ν x. The lack of strict balance between excitation and inhibition in the models with conductance-based synapses indicates that another mechanism is responsible for the highly irregular firing. This can be understood in terms of the large synaptic conductance which is directly reflected in the effective membrane time constant, τ 0, and influences spike-activity in two important ways. First, as the external excitatory drive, ν x, increases, τ 0 decreases so as to keep the equilibrium membrane potential fluctuating around a sub-threshold mean with occasional random excursion to threshold (see Fig.1.A1, C1, and D1 and Eqs. (15), (23)). This is the result of the increased leakiness of the membrane. However, as pointed out by Troyer and Miller (1997) and Bugmann et al. (1997), this effect is insufficient to guarantee highly irregular firing: one also requires that refractory effects are relatively short compared to the mean inter-spike interval, so that their regularizing effect is minimized. The refractory effects include the absolute refractory period and the time taken to recharge to the mean membrane potential from the after-spike reset, V r. In their models a partial V r > V p was used to reduce the time spent in refractory. In the models with conductance-based synapses considered here, the same effect is achieved through relatively short values for τ 0, which controls the time spent recharging to the mean membrane potential after reset. This second effect of τ 0 is illustrated in Fig.3.B1 for the IF model with conductance-based synapses. It shows s = (τ r + τ 0 )ν 0, an estimate of the total time spent in refractory, as a fraction of the mean inter-spike interval, plotted as function of r for four different values of external input, ν = {1, 2, 4, 8} ν th. For values of r corresponding to the irregular firing regime (r > 1), this ratio is much less than 1, indicating that the refractory period is a relatively minor part of the typical inter-spike interval. This is also true of the HH model with conductance-based synapses (Fig.3.B3), but not true for the IF model with current-based synapses, which has significantly large (> 0.1) values of this ratio (Fig.3.B2). Together, these observations imply that there are somewhat different mechanisms underlying the irregular firing in models with conductance-based as opposed to current-based synapses. In the case of conductance-based synapses, irregular firing is maintained across different levels of external input through changes in τ 0

21 (Fig.1.C1) that cause a rapid charging (Fig.3.B1) of the membrane potential to values just below threshold (Fig.1.D1) from which random fluctuations (Fig.1.E1) occasionally bring it over threshold. A strict balance between excitation and inhibition is not required (Fig.3.A1) since regulation of µ 0 is mediated dynamically though τ 0. In the case of current-based synapses, a near balance is required (Fig.3.A2). As the level of external input varies, the large connectivity of cortical cells means that even small deviations from perfect balance cause large changes in the mean membrane potential, possibly to values well below threshold (Fig.1.D2). These large deviations are countered by large values in the variance of the membrane potentials (Fig.1.E2), leading to occasional large transient hyperpolarized membrane potential excursion and consequently values of CV 0 that can be significantly greater than 1 (Fig.1.B2). The two types of model are similar only in as much as the preponderance of recurrent inhibition in the irregular spiking regime dictates that the (free) mean membrane potential be subthreshold; but mechanisms by which this is achieved are quite different, as described above. Similar conclusion were reached in a recent simulation-based study (Rudolph & Destexhe, 2003a). 5.2 Functional implications and future work Recently it has been proposed that the gain of neurons to a stimulus-evoked input can be modulated by the level of balanced background activity (Chance et al., 2002). In this scenario, the overall level of external input increases with stimulus intensity thus driving up the level of background activity, which has the effect of reducing the gain (see (Chance et al., 2002; Burkitt et al., 2003) for a detailed explanation). The proposal relies on random background activity impacting on conductance-based synapses, as studied here. Studies have also assumed that as the level of background activity changes a balance between excitation and inhibition is maintained (Chance et al., 2002; Burkitt et al., 2003; Longtin et al., 2002). This balance is not observed for the recurrent networks with conductance-based synapses studied here (Fig.3.A1 and A3, although this is approximately true for the model with current-based synapses). Nonetheless, the arguments explaining the modulatory effect of background activity on gain are likely to hold under changes in background activity of the sort found here, though their continued validity requires investigation. There has been ongoing debate as to whether cortical neurons act as temporal integrators or coincidence detectors of synaptic input. To some extent this questions