Noise promotes independent control of gamma oscillations and grid firing within recurrent attractor networks

Neural computations underlying cognitive functions require calibration of the strength of excitatory and inhibitory synaptic connections and are associated with modulation of gamma frequency oscillations in network activity. However, principles relating gamma oscillations, synaptic strength and circuit computations are unclear. We address this in attractor network models that account for grid firing and theta-nested gamma oscillations in the medial entorhinal cortex. We show that moderate intrinsic noise massively increases the range of synaptic strengths supporting gamma oscillations and grid computation. With moderate noise, variation in excitatory or inhibitory synaptic strength tunes the amplitude and frequency of gamma activity without disrupting grid firing. This beneficial role for noise results from disruption of epileptic-like network states. Thus, moderate noise promotes independent control of multiplexed firing rate- and gamma-based computational mechanisms. Our results have implications for tuning of normal circuit function and for disorders associated with changes in gamma oscillations and synaptic strength. DOI: http://dx.doi.org/10.7554/eLife.06444.001


Introduction 44
Cognitive processes are mediated by computations in neural circuits and are often 45 associated with gamma frequency oscillations in circuit activity. Gamma activity and 46 cognitive performance often co-vary within tasks and between individuals, while cognitive 47 layer of I cells, which in turn feedback onto the E cell layer ( Figure 1A). For attractor 119 dynamics to emerge the strength of E and I connections are set to depend on the relative 120 locations of neurons in network space ( Figure 1B). While suitable connectivity could arise 121 during development through spike timing-dependent synaptic plasticity (Widloski and 122 Fiete, 2014), here the connection profiles are fixed (Pastoll et al., 2013). To vary the 123 strength of excitatory or inhibitory connections in the network as a whole we scale the 124 strength of all connections relative to a maximum conductance value (g E or g I for excitation 125 and inhibition respectively)( Figure 1B). We also consider networks in which the connection 126 probability, rather than its strength, varies according to the relative position of neurons in 127 the network (Figure 1figure supplement 1). Each E and I cell is implemented as an 128 exponential integrate and fire neuron and so its membrane potential approximates the 129 dynamics of a real neuron, as opposed to models in which synaptic input directly updates 130 a spike rate parameter. Addition of noise to a single E or I cell increases variability in its 131 membrane potential trajectory approximating that seen in vivo ( Figure 1C What happens to grid firing patterns when the strengths of excitatory and / or inhibitory 142 synaptic connections in the model are modified? To address this we first evaluated grid 143 firing while simulating exploration within a circular environment with a network from which 144 noise sources were absent (Figure 2A). When we reduce the strength of connections from 145

I cells by 3-fold and increase the strength of connections from E cells by 3-fold we find that 146
grid firing is abolished (Figure 2Ab vs 2Aa). Exploring the parameter space of g E and g I 147 more systematically reveals a relatively restricted region that supports grid firing ( Figure  148 2D and Figure 2 figure supplement 1A-D). Rather than the required g I and g E being 149 proportional to one another, this region is shifted towards low values of g I and high g E . 150 Thus, the ability of recurrently connected networks to generate grid fields requires specific 151 tuning of synaptic connection strengths. 152 153 Because neural activity in the brain is noisy (Faisal et al., 2008; Shadlen and Newsome, 154 1994), we wanted to know if the ability of the circuit to compute location is affected by 155 noise intrinsic to each neuron ( Figure 1C). Given that continuous attractor networks are 156 often highly sensitive to noise (Eliasmith, 2005;Zhang, 1996), we expected that intrinsic 157 noise would reduce the parameter space in which computation is successful. In contrast, 158 when we added noise with standard deviation of 150 pA to the intrinsic dynamics of each 159 neuron, we found that both configurations from Figure 2Aa,b now supported grid firing 160 patterns (Figure 2Ba,b). When we considered the full space of E and I synaptic strengths 161 in the presence of this moderate noise we now found a much larger region that supports 162 grid firing ( Figure 2E and Figure 2figure supplement 1E-H). This region has a crescent-163 like shape, with arms of relatively high g I and low g E , and low g I and high g E . Thus, while 164 tuning of g I and g E continues to be required for grid firing, moderate noise massively 165 increases the range of g E and g I over which grid fields are generated. 166 167 When intrinsic noise was increased further, to 300 pA, the parameter space that supports 168 grid firing was reduced in line with our initial expectations (Figure 2Ca,b, F and Figure 2 -169 figure supplement 1I-L). To systematically explore the range of g E and g I over which the 170 network is most sensitive to the beneficial effects of noise we subtracted grid scores for 171 simulations with 150 pA noise from scores with deterministic simulations ( Figure 2G). This 172 revealed that the unexpected beneficial effect of noise was primarily in the region of the 173 parameter space where recurrent inhibition was strong. In this region, increasing noise 174 above a threshold led to high grid scores, while further increases in noise progressively 175 impaired grid firing ( Figure 2H). In probabilistically connected networks, the range of g E 176 and g I supporting grid firing was reduced, but the shape of the parameter space and 177 dependence on noise was similar to the standard networks ( Figure 2figure supplement  178 2), indicating that the dependence of grid firing on g E and g I , and the effects of noise, are 179 independent of the detailed implementation of the E-I attractor networks. fields with significant spatial stability, but low spatial sparsity and grid scores compared to 186 excitatory grid cells (Buetfering et al., 2014). A possible interpretation of these data is that 187 parvalbumin positive cells are unlikely to fulfill the roles of I cells predicted in E-I models. 188 However, in networks that we evaluate here in which E cells have grid firing fields in the 189 presence of moderate noise, I cell firing fields also have a much lower spatial information 190 content and spatial sparsity than the corresponding E cell firing fields (E cells: spatial 191 sparsity 0.788 ± 0.061, spatial information: 1.749 ± 0.32 bits/spike; I cells: spatial sparsity 192 0.239 ± 0.018, spatial information 0.243 ± 0.024 bits/spike; p < 10 -16 for comparisons of 193 both spatial sparsity and information; paired t-test; data range is indicated as mean ± 194 standard deviation) ( 1.69 ± 0.18 bits/spike, whereas I cells had grid scores of 0.16 ± 0.2 (p < 10 -16 , paired t-208 test), spatial sparsity of 0.21 ± 0.01 (p < 10 -16 , paired t-test) and spatial information of 0.2 ± 209 0.01 bits/spike (p < 10 -16 , paired t-test; range of all data sets is mean ± SD). Thus, spatial 210 firing of I cells has a similar dependence on noise, g E and g I to grid cells, conventional 211 indices of spatial firing are nevertheless much lower for I cells in E-I networks compared to 212 E cells, and grid firing by E cells in E-I networks is relatively robust to disruption of the 213 rotational symmetry of I cell firing fields. 214

215
To explore gamma activity across a wider range of g I and g E we automated quantification 240 of the strength and frequency of oscillatory input to E cells (see Methods). In the absence 241 of noise gamma frequency activity only occurred for a narrow range of g I and g E (Figure  242 3D). Strikingly, following addition of moderate noise the region of parameter space that 243 supports gamma activity was massively expanded ( Figure 3E). Within this space, the 244 amplitude of gamma increased with increasing inhibition, whereas the frequency was 245 reduced. As noise is increased further the amplitude and frequency of gamma oscillations 246 are reduced ( Figure 3F). We found a similar dependence of gamma oscillations on noise, 247 g E and g I in networks with probabilistic connectivity ( To determine whether there is a systematic relationship between values of g E and g I that 251 generate gamma and grid firing we compared the gridness score and gamma scores for 252 each circuit configuration ( Figure 3G, Figure 3 figure supplements 2 and 3). We found 253 this relationship to be complex and highly sensitive to noise. However, we did not find any 254 evidence for strong linear relationships between gamma amplitude or gamma frequency 255 and grid score (R 2 < 0.12 for all comparisons), while gamma amplitude and frequency 256 provided only modest amounts of information about grid scores (0.27 < MIC < 0.33 and 257 0.27 < MIC < 0.37 respectively). The relationship between noise intensity and gamma 258 differed from that for grid computations. Whereas, grids emerged above a sharp noise 259 threshold ( Figure 2H), for the same regions in parameter space the frequency and 260 amplitude of gamma oscillations varied smoothly as a function of noise ( Figure 3H). Thus, 261 neither the frequency nor the power of gamma appears to be a good predictor of grid firing. 262 263 When we considered only regions of parameter space that generate robust grid fields (grid 264 score > 0.5), we found circuits generating almost the complete observed range of gamma 265 amplitudes (0.02 < autocorrelation peak < 0.59) and frequencies (31 Hz < frequency < 102 266 Hz) (Figure 3figure supplement 4). For example, considering the crescent shaped region 267 of E-I space that supports grid firing in the presence of intermediate noise (the region 268 within the isocline in Figure 3E), when g I is high and g E low then the amplitude of gamma 269 is relatively low and the frequency high. Moving towards the region where g I is high and g E 270 is low, the amplitude of gamma is increased and the frequency is reduced. Thus, variation 271 of synaptic strength across this region of E-I space can be used to tune the properties of 272 gamma activity while maintaining the ability of the network to generate grid fields. 273

274
Together these data indicate that an optimal level of noise promotes emergence of gamma 275 oscillations, while the properties of oscillations may depend on the relative strength of 276 synaptic connections. The relationship between gamma and synaptic strength differs to 277 that for grid computations. Strikingly, while gamma activity provides relatively little 278 information about grid firing, differential sensitivity of gamma and grid firing to g E and g I 279 provides a mechanism for circuits to tune gamma frequency activity while maintaining the 280 ability to compute rate coded grid firing fields. 281 282

Noise promotes attractor computation by opposing seizures 283
Given the emergence of a large parameter space that supports grid firing following 284 introduction of moderate noise, we were interested to understand how noise influences the 285 dynamics of the E-I circuits. One possibility is that in networks that fail to generate grid 286 firing fields network attractor states form, but their activity bumps are unable to track 287 movement. In this scenario disrupted grid firing would reflect incorrect control of network 288 activity by velocity signals. Alternatively, deficits in grid firing may reflect failure of network 289 attractor states to emerge. To distinguish these possibilities we investigated formation of 290 activity bumps in network space over the first 10 s following initialization of each network 291 ( Figure 4). 292

293
Our analysis suggests that the deficit in grid firing in deterministic compared to noisy 294 networks reflects a failure of attractor states to emerge. For deterministic simulation of the 295 points in parameter space considered in Figure 2Aa, which are able to generate grid 296 patterns, we found that a single stable bump of activity emerged over the first 2.5 s of 297 simulated time (Figure 4Aa). In contrast, for deterministic simulation of the point 298 considered in 2Ab, which in deterministic simulations did not generate grid patterns, a 299 single stable bump fails to emerge (Figure 4Ab). Quantification across the wider space of 300 g E and g I values (see Methods) indicated that when g I is low there is a high probability of a 301 bump formation as well as grid firing, whereas when g I is high the probability of both is 302 reduced ( Figure 4B). In contrast to the deterministic condition, for circuits with intrinsically 303 noisy neurons activity bumps emerged in the first 1.25 s following initialization of the 304 network (Figure 4Ac-e) and the area of parameter space that supported bump formation 305 was much larger than that supporting grid firing ( Figure 4B). Plotting gridness scores as a 306 function of bump probability indicated that bump formation was necessary, although not 307 sufficient for grid formation ( Figure 4C), while plotting the first autocorrelation peak as a 308 function of bump probability supported our conclusion that grid computation and gamma 309 activity are not closely related ( Figure 4D). Together, these data indicate that noise 310 promotes formation of attractor bumps in network activity and in deterministic simulations 311 the failure of the circuit to generate attractor states largely accounts for disrupted grid 312

firing. 313
In noisy networks the presence of low grid scores for networks with high bump scores 315 ( Figure 4C) is explained by sensitivity of these network configurations to noise-induced 316 drift. This is illustrated by the region of parameter space from Figure 2Ab, where g I is 317 relatively high and g E relatively low, and which in deterministic simulations fails to generate 318 bumps or grids. With moderate noise, this point generates bumps that show little drift 319 (Figure 4Ac), whereas as noise is increased further the bump begins to drift (Figure 4Ae). 320 In contrast, at the point illustrated in Figure 2Aa  To determine whether these changes in network dynamics are seen across wider regions 348 of parameter space we first quantified the presence of seizure like events from the 349 maximum population firing rate in any 2 ms window over 10 s of simulation time (E-350 rate max ). Strikingly, we found that in the absence of noise epochs with highly synchronized 351 activity were found for almost all combinations of g E and g I , whereas these seizure-like 352 events were absent in simulations where noise was present ( Figure 5D). Interestingly, 353 while grids emerge in deterministic networks in regions of E-I space where E-rate max is 354 relatively low, there is a substantial region of parameter space in which E-rate max is > 400 355 Hz, but grids are nevertheless formed. It is possible that seizure-like states may be rare in 356 this region of parameter space and so do not interfere sufficiently with attractor dynamics 357 to prevent grid firing. To test this we calculated for each combination of g E and g I the 358 proportion of theta cycles having events with population-average rate > 300 Hz (P E-rate > 359 300). For values of g E and g I where grid fields are present P E-rate > 300 was relatively low, 360 indicating that seizure-like events are indeed rare ( Figure 5E). Consistent with this, when 361 we plotted grid score as a function of P E-rate > 300, we found that P E-rate > 300 was 362 relatively informative about the gridness score in networks without noise (MIC = 0.624) and 363 a low value of P E-rate >300 was necessary for grid firing ( Figure 5F). In contrast, E-rate max 364 was less informative of grid firing (0.392 <= MIC <= 0.532) and a wide range of values 365 were consistent with grid firing ( Figure 5F). Thus, while grid firing is compatible with 366 occasional seizure-like events, when seizure-like events occur on the majority of theta 367 cycles then grid firing is prevented. 368 369 Because seizure-like events tend to initiate early on the depolarizing phase of each theta 370 cycle, we asked if synchronization by theta frequency drive plays a role in their initiation. 371 When theta frequency input was replaced with a constant input with the same mean 372 amplitude, the power of gamma oscillations was still dependent on the levels of noise and 373 changes in g E and g I ( Figure 6 figure supplement 1). However, in contrast to simulations 374 with theta frequency input ( Figure 5D,E), noise-free networks without theta exhibited 375 hyper-synchronous firing only when g E was < 0.5 nS ( Figure 6A) and generated grid firing 376 fields almost in the complete range of g E and g I ( Figure 6D high gridness scores (> 0.5) retained its crescent-like shape ( Figure 6E,H), but was 380 smaller when compared to the networks with theta frequency inputs (size of regions with 381 and without theta: 488/961 vs. 438/961), while the range of gamma frequencies present 382 was much lower than in networks containing theta drive. Together, these data indicate that 383 moderate noise prevents emergence of seizure like states by disrupting synchronization of 384 the attractor network by the shared theta frequency drive. In networks with moderate noise 385 theta drive promotes grid firing and enables a wide range of gamma frequencies to be 386 generated without disrupting grid firing. 387 388 Our analysis points towards suppression of seizure-like events as the mechanism by 389 which moderate noise promotes grid firing, while interactions between noise and theta 390 appear important for the capacity to multiplex grid firing with a wide range of gamma 391 frequencies. However, we wanted to know if other factors might contribute to these 392 beneficial roles of noise. Grid fields may also fail to form if overall activity levels are too 393 low, in which case neurons with grid fields instead encode head direction (Bonnevie et al., 394 2013). This observation is unlikely to explain our results as the mean firing rate of E cells 395 in networks that generated grid firing fields (grid score > 0.5, networks with g E or g I set to 0 396 excluded) was in fact lower than the firing rate of networks without grid fields (1.   The distinct control of rate coded grid computations and gamma oscillations by noise, g E 492 and g I was independent of the detailed implementation of the E-I models we considered 493 and was maintained in more complex models incorporating I-I and E-E coupling. Current 494 available experimental data appears to be insufficient to distinguish between these 495 different models. For example, our analysis of interneuron firing indicates that while E-I 496 models predict that interneurons will have spatial firing fields, they have lower spatial 497 information content, spatial sparsity and grid scores than E cells and therefore may be 498 difficult to detect in existing experimental datasets and with current analysis tools. Thus, 499 evidence previously interpreted to argue against E-I based mechanisms for grid firing may 500 in fact not distinguish these from other possible mechanisms. Indeed, we found that grid 501 firing by E cells can be maintained during spatial input that distorts the spatial firing pattern have been difficult to reconcile with rate-coded representations with which they co-exist. 533 We were able to address this issue directly by analyzing a circuit in which gamma 534 oscillations and rate-coded computations arise from a shared mechanism. Rather than 535 gamma serving as an index of rate-coded computation, we find instead that there is a 536 substantial parameter space across which rate-coded computation is stable, while the probabilistically instead in an all-to-all way, the synaptic weights from E to I cells and vice 584 versa were constant, while the probability of connection between the pre-and post-585 synaptic neuron was drawn according to Figure 1B. In addition, some networks also 586 included direct uniform recurrent inhibition between I cells (Figure 7; referred to as E-I-I 587 networks) or direct structured recurrent excitation between E cells (Figure 7figure  588 supplements 6 -10). When recurrent excitation was present, synaptic weights between E 589 cells followed the connectivity profile in which the strongest connection was between cells 590 that were close to each other in network space ( Figure 1B) and the weights between E and 591 I cells were generated either according to synaptic profiles from Figure 1B (Figure 7 -592 figure supplements 6 -9) or the E-I connectivity was uniform with a probability of 593 connection of 0.1 (Figure 7figure supplement 10). E and I cells also received the theta 594 current drive which was the sum of a constant amplitude positive current and a current 595 with sinusoidal waveform (8 Hz). The constant component of the drive was required to 596 activate the circuit, while the theta drive frequency was chosen to reflect the frequency of 597 theta oscillations in behaving animals. The amplitude (cf. Appendix 1) was chosen to 598 produce theta modulation of I cell firing similar to that observed in behaving animals (cf. 599 Chrobak and Buzsaki, 1998) and ex-vivo models of theta-nested gamma activity (cf.       Since the presence of bump attractors is necessary for grid computation, we tested 1205 whether networks with only structured E-E connections can generate activity bumps. We