High-frequency oscillations and replay in a two-population model of hippocampal region CA1

Hippocampal sharp wave/ripple oscillations are a prominent pattern of collective activity, which consists of a strong overall increase of activity with onmodulated (140 – 200 Hz) ripple oscillations. Despite its prominence and its experimentally demonstrated importance for memory consolidation, the mechanisms underlying its generation are to date not understood. Several models assume that recurrent networks of inhibitory cells alone can explain the generation and main characteristics of the ripple oscillations. Recent experiments, however, indicate that in addition to inhibitory basket cells, the pattern requires in vivo the activity of the local population of excitatory pyramidal cells. Here we study a model for networks in the hippocampal region CA1 incorporating such a local excitatory population of pyramidal neurons and investigate its ability to generate ripple oscillations using extensive simulations. We find that with biologically plausible values for single neuron, synapse and connectivity parameters, random connectivity and absent strong feedforward drive to the inhibitory population, oscillation patterns similar to in vivo sharp wave/ripples can only be generated if excitatory cell spiking is triggered by short pulses of external excitation. Specifically, whereas temporally broad excitation can lead to high-frequency oscillations in the ripple range, sparse pyramidal cell activity is only obtained with pulse-like external CA3 excitation. Further simulations indicate that such short pulses could originate from dendritic spikes in the apical or basal dendrites of CA1 pyramidal cells, which are triggered by coincident spike arrivals from hippocampal region CA3. Finally we show that replay of sequences by pyramidal neurons and ripple oscillations can arise intrinsically in CA1 due to structured connectivity that gives rise to alternating excitatory pulse and inhibitory gap coding; the latter implies phases of silence in specific basket cell groups and selective disinhibition of groups of pyramidal neurons. This general mechanism for sequence generation leads to sparse pyramidal cell and dense basket cell spiking, does not rely on synfire chain-like feedforward excitation and may be relevant for other brain regions as well. Author summary During certain phases of sleep, rest and consummatory behavior the hippocampus brain area of many species, including humans, is known to intermittently generate strong high frequency oscillations. These oscillations are important for memory formation and consolidation. To date, the mechanisms underlying their generation remain incompletely understood. We find that in unstructured networks carefully designing how excitation is transmitted in the hippocampus is required for the generation of robust fast oscillations in its main output region. Broad, temporally extended excitation of cells results in unrealistic single cell activity, whereas temporally narrow input that differs from cell to cell gives rise to oscillations with realistic single cell and network activity. We show that the biophysical mechanism to generate the required temporally narrow excitation may be related to spiking events in the dendrites, which are triggered by coincident input. Our results in structured networks suggest that the interplay of hippocampal excitation and inhibition can serve as a means to generate robust sequential activity, which is thought to be crucial for memory formation and recall. The sequence generation mechanism also leads to strong high frequency oscillations with sparse excitatory cell and frequent inhibitory cell spiking, as observed in the hippocampus.

Hippocampal sharp wave/ripples (SPW/Rs) are remarkable both from a 2 neurophysiological viewpoint and in their behavioral impact: On the one hand, they 3 consist of strong increases of spiking activity in large parts of local neuron populations 4 (the sharp wave) together with oscillatory, extraordinarily coherent neuronal discharges 5 (ripples); on the other hand, SPW/Rs have directly been shown to be important for 6 memory consolidation [1][2][3][4][5][6] and might be involved in planning of future actions [1,7]. 7 Perhaps the most striking signature of memory consolidation is the phenomenon of 8 hippocampal replay (see [8] for a recent review) during which sequences of pyramidal 9 cell action potentials encoding location, so-called place cells [9], are repeated on a faster 10 timescale during SPW/R complexes. 11 SPW/Rs occur in all mammalian species that have been investigated for them, 12 including humans [1,10]. Similar activity has been observed in fish [1], reptiles [11] and 13 birds [12], indicating that the brain circuits involved in their generation are evolutionary 14 old. The existence of SPW/Rs in non-mammalian species remains, however, a 15 controversial subject [1,12]. 16 Our understanding of the role of SPW/Rs in the brain would benefit from a detailed 17 understanding of the underlying generative mechanism, which includes the external during hippocampal replay, can be generated intrinsically in CA1 by a mechanism 144 involving disinhibition of selected groups of pyramidal cells. 145 Materials and methods 146 We consider three variants, model 1, 2 and 3, of a two-population model consisting of 147 N E excitatory pyramidal cells (PCs), the principal cells of hippocampal region CA1, 148 and N I inhibitory parvalbumin-positive BCs (Fig 1). 149

Fig 1. Model overview.
Schematic drawing of the three models considered in this paper. They consist of a population of N E excitatory and a population of N I inhibitory CA1 neurons, which are connected with probabilities p αβ , α, β ∈ {E, I} (lower panel) and receive external input from CA3 (upper panel). The models differ in this external input: In model 1 and 2, it is represented by the excitatory conductances g ext (Eq 6) that it evokes in n E CA1 neurons. In model 1, the time courses of these conductances are broad and for each neuron centered around the same time (upper left panel). In model 2, the time courses are short, pulse-like and centered around different times (upper middle panel). In model 3, CA3 is represented by a population of input neurons. Each such neuron fires according to an inhomogeneous Poisson process with Gaussian rate profile given by Eq 7 (upper right panel). Spike transmission from CA3 to CA1 is then filtered by a strong apical or basal dendrite of the receiving CA1 neuron: Besides the linear input transmission, sufficiently coincident input evokes dendritic spikes. These generate somatic currents whose peak strengths follow a lognormal distribution (Eq 2) across neurons.
Their dynamics are governed by a conductance-based integrate-and-fire neuron The leak current is given by I E leak = g E l E E rest − V (t) and the excitatory and inhibitory 152 currents are given by , 153 respectively. The total excitatory conductance consists of two parts: 154 g E tot (t) = g E exc (t) + γg ext (t), representing input from recurrent excitatory synapses and 155 external input. The external input is a direct, not dendritically amplified, input present 156 in model 1 and 2. It models input from CA3 or an optogenetic stimulation delivered in 157 experiments [22]. γ ∈ {0, 1} determines whether this drive is present in the model, i.e. 158 γ = 1 in models 1 and 2, whereas γ = 0 in model 3. The noise current 159 I E noise = σ n 2g E l C E ξ(t) is a Gaussian white noise input, independent between neurons 160 and with strength σ n = 1 mV. 161 I dendritic is a somatic current triggered by apical or basal dendritic spikes and only 162 present in model 3. We implement the generation of dendritic spikes via a temporal 163 coincidence detection mechanism [43,44]: As soon as the dendrite receives sufficiently 164 many spikes from its presynaptic CA3 neurons within the dendritic integration window 165 w D = 2 ms (< 3 ms [65,66]), a fast and strong, but not necessarily suprathreshold 166 excitatory current I dendritic is generated. It depolarizes the somatic membrane potential 167 a delay τ D = 2 ms after the coincidence was detected. There is a refractory period 168 t r,D = 5.0 ms during which no further dendritic spikes are triggered -this means that 169 after a dendritic spike, there cannot be another one for a duration of t r,D . We checked 170 that our results do not depend critically on the exact value of t r,D by performing 171 simulations with smaller (t r,D = 2 ms) and larger (t r,D = 10 ms) values. The results 172 that we will report below for model 3 remained qualitatively unchanged, only the 173 parameters of the lognormal distribution (Eq 2) where HFOs in the ripple range occur 174 shifted such that a smaller (larger) t r,D was compensated for by an increase (decrease) 175 in µ and/or σ. 176 We assume that every pyramidal neuron has one dendritic compartment where it 177 receives the relevant input from CA3. This models the single main apical dendrite in 178 CA1 pyramidal neurons [15,65,67] and its several oblique ones emanating from it. We 179 do not explicitly model the compartment in this study, but instead focus on the impact 180 a dendritic spike has on the somatic membrane voltage. All inputs from CA3 contribute 181 in the same way to dendritic spike generation. In particular we do not separately model 182 local effects such as the generation of weak dendritic spikes in the oblique dendrites [68]. 183 Because innervation by CA3 Schaffer collaterals occurs on both basal and apical 184 dendrites of CA1 pyramidal neurons [15], the modeled dendrite may alternatively be 185 interpreted as an influential, privileged basal one [69] (neglecting the contribution of 186 recurrent CA1 excitatory inputs to dendritic spike generation in such a dendrite -187 dendritic spikes in our model 3 are generated by feedforward Schaffer collateral input). 188 Experiments show that the impact of dendritic spikes on the soma can be different 189 depending on the generating dendrite, but also on the depolarization and the somatic 190 firing history [65,70,71]. If not mentioned otherwise we thus assume that the strengths 191 of dendritic spikes across neurons have a lognormal distribution as experimentally found 192 for several other neuronal properties [60,72]. Specifically the peaks of the currents 193 induced in the soma have a lognormal distribution across neurons, 194 I peak dendritic ∼ lognormal(µ, σ) .
The current in the soma induced by the dendritic spike has a rise time of 1 ms and a of the lognormal distribution, which yields 203 CDF(x; µ, σ) = 0.9983 with x = 10 nA, µ = 0 nA and σ = 0.75 nA. We note that 204 somatic action potential bursts in CA1 pyramidal cells are usually generated by a more 205 involved mechanism, where calcium spikes in the apical tuft are generated with the 206 support of backpropagating action potentials [73,74]. A dendritic spike is generated at t = 12 ms. It is modeled by the dendritic current I dendritic (lower subpanels in A,B,C) arriving at the soma τ D = 2 ms later (here at 14 ms). For small peak dendritic currents (I peak dendritic = 0.1 nA, panel A), the somatic depolarization (upper subpanel) is small (here: 1.28 mV) when starting at rest. For larger peak dendritic currents, the somatic depolarization increases (12.78 mV for I peak dendritic = 1 nA, panel B), but remains subthreshold. For I peak dendritic ≥ 1.34 nA, at least one somatic spike is generated; for I peak dendritic = 5 nA (panel C), two spikes are generated and there is significant depolarization visible in the membrane voltage even after the second spike. The scale for the bottom panels is fixed to facilitate comparison between the different values for the dendritic peak current.
The dynamics of the inhibitory neurons are given by similar to Eq 1. The leak, excitatory and inhibitory currents are respectively. I I noise = σ n 2C I g I l ζ(t) is a Gaussian white noise input, independent 212 between neurons and with strength σ n = 1 mV. In both the E and I populations, once a 213 neuron reaches its firing threshold V α thresh (α ∈ {E, I}), it is reset to V α reset and remains 214 at this potential during an absolute refractory period t α ref .

215
The neurons are connected by chemical synapses, see next section for details. Gap 216 junctions between inhibitory PV+BCs in CA1 are frequent [47,75], but their 217 importance for ripple oscillations in vivo is unclear [76,77]. In the context of IR models, 218 recent theoretical results show that adding gap junctions between inhibitory cells has 219 beneficial effects for ripple oscillations, as they enhance synchrony and decrease the 220 minimal number of cells required for a ripple oscillation [78]. Since our models do not 221 require gap junctions and because of their unclear importance for SPW/Rs, we do not 222 consider them in this paper. 223 We have chosen the single neuron and network parameters in agreement with 224 neuroanatomical and neurophysiological experimental knowledge on the hippocampal 225 area CA1. The data we refer to come mostly from rats, but also from mice [57]. The 226 parameters can be found in the Supporting Information (SI), section Parameters.

227
Synaptic dynamics and connectivity 228 The excitatory and inhibitory synaptic conductances induced by a presynaptic spike at 229 time t = 0 are given by a bi-exponential function, for t ≥ τ l , where τ l is the transmission delay, and g α β (t) = 0 otherwise. s α β is a constant 231 ensuring that g α β has its maximum at g α β,peak . Suppressing the indices α and β we have 232 This setup is similar to that used in [44] and [25]. We show time courses for post 233 synaptic potentials, conductances and post-synaptic currents in Fig 11, in the SI section 234 Synaptic dynamics in CA1, for a fixed holding potential of −55 mV.

235
The two neuronal populations are coupled uniformly at random, with probabilities 236 p αβ , where α ∈ {E, I} denotes the presynaptic population and β ∈ {E, I} denotes the 237 postsynaptic population. If not mentioned otherwise, we set N E = 12000 and N I = 200, 238 which approximately agrees with the neuron numbers in a typical CA1 slice, of thickness 239 0.4 mm and a volume of 0.057 mm 3 [25]. The numbers respect the ratio of pyramidal 240 cells to PV+ basket cells in rat CA1, which is approximately 60:1 [57]. To compute the 241 connection probabilities p αβ , we assume that all connections between E and I cells are 242 realized in the subset of CA1 neurons we consider. This means that we first determine 243 how many inputs each E or I cell receives from the other cells in the population and then 244 compute the connection probabilities using the numbers for N E and N I given above.

245
A hallmark of CA1 is its very sparse recurrent excitatory connectivity [56,57]. We 246 set p EE = 1.64% [56] so that for N E = 12000, every pyramidal cell receives input from 247 approximately 200 other pyramidal cells.

248
For the I-to-I connectivity, we set p II = 0.2 [57,79,80], which, for N I = 200, means 249 that each basket cell on average receives 40 synapses from other PV+BCs in the network. 250 This value is obtained as follows: a single PV+BC contacts on average 64 other PV+ 251 cells [79], of which 60% are basket cells [57], so that a single PV+BC contacts 38 other 252 PV+BCs on average [25]. This corresponds approximately to p II = 0.2.

253
For the I-to-E synapses, we obtain several estimates, (i) based on the experimentally 254 observed number of PCs that are postsynaptic to a single PV+BC and (ii) based on the 255 experimentally observed number of PV+BCs that are presynaptic to a single PC. We 256 first consider ref. [ p EI = 0.1 in this study. This is at the upper end of the biologically plausible values, but 287 acceptable because we will systematically vary the excitatory drive to the E population 288 in our simulations: Decreasing p EI is similar to decreasing the number of active E cells 289 since the latter reduces the number of the realized E-to-I connections that will have a 290 postsynaptic effect. A decrease in the number of active E cells results in our models 291 from changing the number n E of PCs that receive CA3 drive in model 1 and 2 and by 292 changing the strength of the CA3 drive in model 3, see next section.

293
In summary, using the values available in the experimental literature [57,79,[81][82][83][84] 294 we obtained for our networks p EI = 0.1, so that each PV+BC receives on average input 295 from 1200 PCs, and p IE = 0.1, so that each PC receives on average input from 20 296 PV+BCs. These connection probabilities are in good agreement with many previous 297 computational studies of CA1 [36,43,44]. Notably, [25] use the same values for the 298 connection probabilities except for a higher value of p IE = 0.3. Our more detailed 299 choice for the connection probabilities differs from that in ref. [37], where a single value 300 of p = 0.2 for all synapses was assumed. Finally, it differs from the all-to-all 301 connectivity for all synapses except the E-to-E synapses assumed by [29].

302
To conclude, as a result of the considerations in this section we set 303 p EE = 0.0164, p II = 0.2, p IE = 0.1 and p EI = 0.1 (if not mentioned otherwise).

Excitation of pyramidal cells by CA3
305 Models 1 and 2 explicitly mimic sharp-wave input from CA3 without modeling 306 CA3 [25]: a subset n E of the pyramidal cells is driven by time-dependent conductances 307 g ext with a Gaussian profile, In model 3, CA3 is modeled as a population of N E,CA3 = 15000 excitatory neurons that 309 each spike according to an inhomogeneous Poisson process with rate time course The connection probability between CA3 and CA1 is p = 130 N E,CA3 ≈ 1% if not mentioned 311 otherwise. This means that every CA1 pyramidal neuron on average receives input from 312 130 CA3 pyramidal neurons. The number is comparable to the typical number of inputs 313 a CA1 neuron receives from CA3 during an SPW/R event (150 − 300 neurons, [85]).

314
This is much lower than the number of CA3 neurons converging on a single CA1 315 pyramidal neuron (15000 − 30000, [57]). However, only approximately 1% of all CA3 316 pyramidal neurons are active during a SPW/R [59,63]. Assuming that rat CA3 contains 317 approximately 205000 PCs [57] and that the average connection probability from CA3 to 318 CA1 (neglecting CA3 sublayer specificity) is between 1 and 8% [57], we obtain between 319 0.01 · 0.01 · 205000 ≈ 21 and 0.01 · 0.08 · 205000 = 164 inputs per CA1 PC, which is 320 consistent with our choice for p given above. In light of this estimate, we consider one 321 hundred to a few hundreds, but not thousand, inputs to a CA1 PC from CA3 to be 322 realistic estimates. Our results do not depend crucially on the exact size of p. For 323 example, a modest increase in p can be compensated for by a decrease in the average 324 strengths of the peak dendritic currents (given by Eq 2) or the peak rate r 0 in eq. 7.

325
As already mentioned above when introducing Eq 1, the impact of temporally 326 coincident spikes from CA3 is nonlinearly amplified, in all N E excitatory cells: if in one 327 of these cells 5 or more spikes arrive in an interval of w D = 2 ms [65], a dendritic spike 328 is triggered. Irrespective of the coincidence detection mechanism, each CA3 spike 329 impinging on a CA1 PC also causes a small depolarization (rise/decay time 1/2 ms, 330 peak conductance 0.75 nA). The dendritic spike is incorporated in the model by the 331 current I dendritic (t) that it generates in the soma (see Fig 2). We assume that the 332 impact of the dendritic spike is stereotypical, i.e. I peak dendritic (cf. Eq 2) is for a given 333 neuron constant over time and independent of the CA3 inputs that triggered it. This is 334 plausible since dendritic spikes are stereotypical and couplings between dendrites and 335 soma are reliable [71]. There are thus two sources of heterogeneity in the excitatory 336 connections from CA3 to CA1. The first source is that the number of CA3 inputs 337 impinging on a given CA1 cell is variable as only the connection probability p (see 338 above) is fixed. The second source is the variable peak dendritic current I peak dendritic 339 (cf. Eq 2) which is drawn, for each CA1 PC independently, from a lognormal 340 distribution (Eq 2) with fixed parameters µ and σ. In our model, not every dendritic 341 spike causes a somatic spike or even has a discernible influence on the soma (see Fig 2), 342 which is in agreement with experimental observations [86]. Dendritic spikes have been 343 observed in the apical dendrites of CA1 PCs during SPW/R [86]. These large-amplitude 344 fast spikes [86] are initially fast and then broad, so that they are likely composed of 345 both a sodium-dependent and a calcium-dependent component, which is generally less 346 sensitive to synchronous coincident spiking [87], see also [88] for an example of a 347 dendritic spike composed both of a calcium and sodium-dependent component. This is 348 a caveat, because it might be that the large-amplitude dendritic spikes in [86] are less 349 sensitive to synchrony in the input from CA3 or not caused by CA3 input at all, but by 350 other mechanisms such as backpropagating action potentials [89]. Assuming that spikes 351 are generated in the apical dendrites of CA1 pyramidal neurons during SPW/R by a 352 coincidence-based mechanism, we can estimate the number of inputs n D required for its 353 generation. To this end, we first define the dendritic coincidence rate as the number of 354 inputs n D that have to arrive in the time window w D to cause a dendritic spike. To 355 generate a dendritic spike, the mean CA3 afferent rate, which is given by r 0 pN E,CA3 , 356 should be at least equal to this event rate, thus we have the equality With w D = 2 ms, pN E,CA3 = 130 and 8 Hz ≤ r 0 ≤ 10 Hz, this gives For w D = 3 ms, the estimate becomes 3 < n D ≤ 3.9. If additionally, pN E,CA3 = 150, we 359 have 3.6 ≤ n D ≤ 4.5. This is lower than n D = 5 that we have assumed above, but on 360 the same order of magnitude. This number of inputs n D and the time window w D are in 361 agreement with the conditions for the generation of spikes in apical dendrites a realistic 362 model of CA1 PCs [90]. Hence, in our model, few, but strong, CA3-CA1 synapses cause 363 dendritic spiking. Our choice of parameters for the CA3-CA1 connectivity and the CA3 364 peak rate also entail that only neurons that receive more than average (r 0 pN E,CA3 ) 365 CA3 input spikes will generate dendritic spikes. If not mentioned otherwise, there is no 366 cutoff in the distribution of the peak dendritic currents. If a cutoff is imposed, after 367 drawing the peak dendritic current for each neuron, all peak dendritic currents above 368 the cutoff are set to 0 nA. We show a schematic summary of our three models in Fig 1. 369 All simulations were performed with custom scripts in brian2 [91].

371
In the following, we study the three minimal CA1 two-population models Fig   Following [22], where it was shown that exciting a small number of CA1 pyramidal 388 cells is necessary and (in the presence of intact inhibition) also sufficient to induce a 389 ripple oscillation, we first do not include any external excitatory drive to the inhibitory 390 population of PV+BCs. That inhibition is also necessary [22] suggests a model of 391 rhythmogenesis where the excited pyramidal cell subpopulation excites inhibitory cells, 392 which then inhibit their excitatory targets. In such a model, the network frequency will 393 depend on the recurrent E-to-I and I-to-E connections [37].

Temporally homogeneous excitation of pyramidal cells (model 1) 395
We first study a model in which a subset of the PC population receives input that peaks 396 at the same time t 0 , but has a different amplitude for each cell. The setup is similar to 397 the indirect drive condition in [25]. In a minimal variant of the network, there is no 398 external excitation of the PV+BC population, that is, the PV+BCs would be silent 399 without input from the PC population. A subset of n E PCs is each excited by a 400 time-dependent conductance (Eq 6), which peaks at t 0 = 50 ms. The amplitudes g 0 ext of 401 the conductances are distributed according to a Gaussian with meanḡ and standard 402 deviation CV gḡ , where CV g is the coefficient of variation, which is kept constant when 403 we varyḡ.  To confirm that this is a generic problem of model 1 and not due to our choice ofḡ 415 and n E , we systematically varied these parameters. The results are shown in Fig. 4.     small (lower panels in A). Networks with parameters that fall into both regions can be 421 expected to generate robust ripple frequency oscillations. However, in the entire region 422 where the expected oscillation frequency is in the ripple frequency range, an active 423 pyramidal cell spikes on average more than 5 times during the whole ripple cycle ( Fig. 4 424 B , Fig 3 B). This is in marked disagreement with experimental findings [20], which show 425 that most pyramidal cells fire once or twice during the ripple cycle. More recent 426 experiments even observed that E cells typically spike only once per ripple [22]. The 427 white horizontal line in Fig 4A roughly delimits the region where E cells spike sparsely. 428 Anywhere below this white line, i.e. for higher values ofḡ, C E is larger than two, which 429 is biologically implausible.

430
Thus, in our model 1, in the absence of external feedforward excitatory drive to 431 inhibitory cells, it is not possible to reach the ripple frequency range with sparse firing 432 of pyramidal cells by changingḡ and n E . This remains true if p IE is increased from 0.1 433 to 0.2 ( Fig. 12, SI, section Higher I-to-E, I-to-I connectivity and broader sharp waves in 434 model 1). We increased p IE because one might expect that more inhibitory inputs   Each E cell receives an input pulse whose duration is comparable to the interval between two ripple waves. The peak times of the pulses are distributed. E and I populations spike at ripple frequency, active E cells typically contribute one to two spikes to an event. A: Input pulses, spike rastergrams and network activities, displayed as in    sharp-wave activity [86]. Ref. [67] suggested that dendritic spikes observed in the apical 523 dendrites of CA1 pyramidal cells endow the cell with unique information processing 524 capabilities during SPW/Rs. Specifically it was shown that these dendritic spikes result 525 in precise somatic spikes with low temporal jitter. This lead to the conjecture that 526 dendritic compartments receiving input clustered both in space and time perform 527 supra-linear dendritic integration during SPW/R events. The dendrites might then act 528 as feature detectors on the CA3 input and determine the neuronal action potential 529 output. This would be largely independent of the mean input strength in contrast to 530 the output during theta oscillations [67].

531
Consistent with these findings and suggestions, we propose that dendritic spikes may 532 provide the short-term excitation that we found to be necessary for HFOs with sparse E 533 cell firing in model 2 (see Fig 6 and Fig 17). For this, we model the generation of 534 dendritic spikes in the E neurons that receive input from CA3. Since dendritic spikes are 535 generated by coincident spike arrivals, we introduce a simple model of CA3 with N E,CA3 536 pyramidal cells, each of which fires according to an inhomogeneous Poisson process.

537
This has a peaked rate profile with the same amplitude and width across all CA3 538 neurons (see Eq 7, the peak is at t 0 = 50 ms, width is σ t = 10 ms). When there are at 539 least five input spikes arriving within a time window of 2ms, a dendritic spike is elicited. 540 Dendritic spikes impact a neuron's somatic voltage in a stereotypical fashion. Between 541 neurons, their impact varies: the peak current elicited by a dendritic spike in the soma 542 is distributed according to a lognormal distribution (Eq 2) across the CA1 E cells. Each 543 CA1 PC receives inputs from a small fraction of all CA3 pyramidal cells. We set the intuitively explained as follows: The spike rate of the CA3 population (given by Eq 7) is 553 towards its peak sufficiently high to generate widespread spiking in the CA1 pyramidal 554 cells. Fast recurrent inhibition within CA1 is required to terminate a single pyramidal 555 ripple wave; in the absence of inhibition it would go on as long as CA3 spiking is strong 556 enough. Inhibition has to impinge early enough on the excitatory population on every 557 ripple wave to prevent an excess of excitation and thus a slowing of the oscillation due 558 to resulting excessive inhibitory spiking. Therefore, a parameter change that lets 559 recurrent inhibition arrive earlier decreases the number of E spikes and renders the 560 oscillations more pronounced and robust. Any mechanism that decreases the width of 561 the excitatory ripple wave will have a similar effect. In addition to the PV+BCs 562 considered here, such silencing of pyramidal cells could be provided by bistratified 563 cells [97,98] or feedforward inhibition [99]. number of spikes in the active E cell population is between 1 and 2 (approximately 1.5 567 in Fig 7) and most active E cells spike once or twice, a small fraction of neurons spikes 5 568 times or more during the SPW/R event (left histogram in Fig 7 B). Such 'bursting' 569 behavior is caused by large values for the peak of the dendritic current and/or 570 exceptionally high synchrony in the presynaptic spike trains impinging on a CA1 PC. 571 We will see below that it is not crucial for generating SPW/Rs in our models (SI, 572 Figs 25,26,27 and 28). Bursting of pyramidal cells in CA1 was observed in vivo during 573 slow wave sleep [60,100], in particular during SPW/Rs [39,101]. considerably across different network realization (Fig 8 A lower panels).

592
E cell spiking is sparse all over the range where ripple frequency oscillations are 593 generated: C E is approximately constant, between 1 and 2 (Fig 8 B). The amount of I 594 cell firing depends on the precise values for the parameters of the peak dendritic current. 595 For negative values of µ and σ ≈ 1 nA, I cells fire on average three times per ripple.

596
Moving along the diagonal where ripple oscillations are generated to values of 597 σ ≈ 0.5 nA and µ ≈ 0.9 nA, increases C I to more than 10 spikes per ripple. The range 598 in between appears consistent with the experimental observation that CA1 PV+BCs 599 typically spike on every ripple wave. 600 We now discuss the impact of certain selected parameter changes on model 3 to firing is sparse within the band. Thus, model 3 is robust to a higher I-to-E connection 607 probability. We argue in Materials and methods that p IE should be between 0.1 and 0.2 608 for the network sizes we consider in this paper. Given that HFOs in the ripple range are 609 still observed with a higher value p IE = 0.3, we are confident that the results of model 3 610 do not hinge critically on the precise value of the I-to-E connection probability. 611 We next increase the I-to-E synaptic latency from 0.5 ms to 0.9 ms (SI, Fig 22). The 612 maximal frequency reached by the E population is reduced to approximately 140 Hz 613 and the region where this occurs is shifted to smaller values of σ. E cell firing remains 614 sparse. This indicates that in model 3 HFOs in the ripple range (frequencies > 160 Hz) 615 rely on fast synaptic transmission from the I to the E population. 616 We also decreased the peak E-to-I conductance g I exc,peak from 3 to 1 nS still in the 617 experimentally observed range [83,84]. This results in more robust oscillations in the  Fig 24). We note that additionally 628 the peak E-to-I synaptic conductance was decreased (as in SI, Fig 23). The changes do 629 not impact the ability of model 3 to generate HFOs in the ripple range with sparse 630 firing of E cells. We conclude that model 3 does not critically hinge on the smaller value 631 for the I-to-E synaptic latency used in Fig 8.

632
Finally, we study whether the large values at the tail of the lognormal distribution of 633 dendritic spike impacts are important. We thus truncate the distribution given by Eq 2 634 such that values of I peak dendritic larger than 4 nA are mapped to 0. With this truncation, a 635 single dendritic spike can generate at most, but barely, two somatic ones (see Fig 2 C 636 for an example showing two somatic spikes as a response to one dendritic spike with a 637 peak amplitude of 5 nA). Thus, with this truncation, there can be no somatic bursting 638 due to a single dendritic spike. We find that this reduces the maximal frequency that 639 can be reached (SI, section Higher p IE , truncation of distribution for I peak dendritic , higher 640 I-to-E synaptic latency and lower E-to-I peak conductance in model 3, Fig 25).

641
However, frequencies around 170 Hz can still be reached and the standard deviation of 642 the frequencies is low in that region, albeit higher than in Fig 8. Additionally, we found 643 that the low ripple range (∼ 150 − 160 Hz) can still be reached when the truncation is 644 introduced at 3 nA, but not when it is introduced at 2 nA (not shown). 645 We additionally study a modified way to truncate the distribution for the peak  without structured input from CA3 [39]. Further experimental evidence indicates that 673 place cells in CA1 and CA3 have different properties [105,107]. One possible explanation 674 for the intrinsic replay in CA1 is that it is based on recurrent excitatory connections 675 that are amplified due to dendritic spikes in the basal dendrites of CA1 [43,44]. In this 676 model, pulses of excitatory spikes travel along pathways in the sparse recurrent 677 connectivity, which is enabled by amplification by dendritic spikes in the basal dendrites. 678 Replay of spike sequences might also arise from continuous attractor dynamics. Here a 679 localized bump of activity in the neural tissue moves around because of asymmetric 680 synaptic connections, short-term plasticity, or adaptation mechanisms [108][109][110][111][112].

681
Sequential network structures might guide these bumps to replay certain sequences.

682
Due to the very sparse excitatory recurrent connectivity, also propagation of inhibitory 683 spikes pulses in an essentially inhibitory network as suggested for the striatum [113], 684 might be considered biologically plausible for CA1. However, given the high firing rate 685 of I neurons during ripples (as mentioned before, PC+BCs fire on nearly every ripple 686 wave [1,20]), it seems unlikely that their population activity forms sequential patterns 687 similar to that of PCs. In support of this, it is also known that I cells have broader, 688 more unspecific place fields than E cells [85,99] (see however [114] and [115]).

689
Motivated by the high sparseness of excitatory-excitatory connectivity in CA1 and 690 the supposed absence of replay in inhibitory activity, we propose an alternative model 691 for how specific sequences may be stored and replayed. The networks are a modification 692 of model 3, in particular there are dendritic spikes amplifying the input from CA3 to 693 CA1. We expect that these are not needed, i.e. any form of strong feedforward 694 excitation on CA1 PCs (for example the form used in model 1) should be sufficient for 695 our replay model. Like the proposed ripple generation mechanism, sequence replay is 696 based on the prominent excitatory-inhibitory and inhibitory-excitatory connectivity in 697 the hippocampal region CA1. The basic idea is as follows: CA3 excites an initial group 698 of CA1 E cells to spike within a short time interval [116]. This group excites all CA1 I 699 cells involved in the dynamical pattern ('pulse coding'), except for a group that would 700 inhibit the second group of E cells. Due to the resulting disinhibition ('gap coding'), 701 this second group of E cells becomes active and excites all I cells except for a group, 702 which would inhibit the third group of E cells. Therefore this group becomes active and 703 so on (Fig 9 A). A generalization to continuous, non-grouped activity sequences [31,44] 704 seems straightforward. Further alternative concepts for replay, which we do not consider 705 as they seem less biologically plausible for CA1, are presented in the SI. The first such 706 concept (Fig 30 A) is a two-population model, which assumes that in addition to E cells 707 also the I cells spike in a sequential fashion: the active group of E cells excites the next 708 group of I cells, which inhibits all E cells except those that fire in the next step. The 709 second concept is an inhibition-first model, which generates sequences by pure gap 710 coding in the I population (Fig 30 B): groups of I cells become sequentially inactive, 711 since they receive at some point more inhibition than their peers because of structured 712 inhibitory connectivity. The resulting disinhibition of E cells leads to their sequential 713 activity, but their spiking does not contribute to maintain the sequence. The sequence replay is based on subsequent disinhibition of a group of E cells and the resulting excitation of all but one group of I cells, which in turn disinhibits another group of E cells. A: Connectivity scheme. Both the E cells and the I cells are grouped into K + 1 groups (here: K = 4 was chosen for simplicity, in the simulations below, K = 9). Group E 1 excites all groups of I cells except I 2 (green dashed line). Similarly, group E k excites all groups I k , but not I k+1 . Group I k projects on a single group E k for all k = 0, ..., K − 1, that is, group I 0 inhibits group E 0 , group I 1 inhibits group E 1 , and so on. All connections relevant for the sequence generation are only shown for group E 1 and I 1 . I-to-I connections are not shown. A ripple event is triggered by the initial stimulation of group E 0 and the ripple event progresses in the direction of the black arrow. The subsequent activation by CA3 lets E neurons spike when they are not inhibited. E 0 activates all I neurons except I 1 . Since I 1 inhibits all E cells except group E 1 , these become active due to the random, unstructured CA3 input. This excites all I cells except those forming group I 2 . The resulting gap in inhibition generates a pulse in E cell group E 2 on the next ripple wave etc. B: Example realization in a network with N E = 5000 excitatory cells and N I = 400 inhibitory cells generating replay together with an oscillation of frequency f ≈ 190 Hz. Group E 0 receives input from the purple CA3 population consisting of 12000 neurons. The remaining groups of excitatory cells receive unstructured input from N E,CA3 = 15000 different excitatory CA3 spike trains shown in black. About seven E (I) groups are sequentially (de)activated, before the sequence terminates due to the termination of CA3 input. Parameters as in Fig 8, except those listed in SI section Parameters of the replay model. peak inhibitory synaptic conductance on E cells is increased so that at the same holding 721 potential as previously, each inhibitory spike now generates a hyperpolarization of 722 1.35 mv instead of 0.83 mV. This modification is biologically plausible given recent in 723 vivo experiments [30], but also consistent with older studies [84]. Further, we chose the 724 parameters of the dendritic spike strength distribution such that E cell spiking can be 725 effectively suppressed by I cell spiking, i.e. we avoid large values for the peak dendritic 726 current. Concretely we set the mean of the peak dendritic current distribution slightly 727 above the value required for the generation of one spike from rest (see Fig 2), and we 728 truncate the distribution at 4 nA as described above.

729
As in Fig 8, the I-to-E synaptic latency is kept at 0.5 ms. This comparably small 730 value is helpful for robust replay accompanied by HFOs in the ripple range: a 731 simulation with increased latency is shown in the SI, Fig 29;

736
In the current article, we have studied two-population models for the generation of 737 ripples and sequences in the hippocampal region CA1. In these models E and I neurons 738 interact to generate the ripple rhythm possibly together with sequential activity. This is 739 motivated by recent in vivo experiments, which have shown that both the local PC and 740 the local PV+BC populations in CA1 contribute to generating ripple oscillations [22]. 741 Further motivation comes from the observation that the connectivity from local PCs to 742 PC+BCs and back is high. Our models are constrained by biologically plausible 743 connectivity and by the fact that during ripple oscillations, basically all PV+BCs spike 744 at high frequency, i.e. on nearly every ripple wave, while the PCs spike sparsely, 745 contributing typically only one or two spikes. 746 We observe that two-population models receiving temporally broad CA3 sharp wave 747 input either generate ripple frequency oscillations or sparse spiking of E neurons, but 748 not both. Sparse spiking of E neurons can be reached if there is a strong CA3 749 projection to the I neurons, such that the oscillations are effectively generated by the I 750 population alone, which seems inconsistent with experimental findings [22]. We have 751 thus explored in more depth another idea, namely that the sparse spiking of E cells in 752 CA1 originates from temporally sparse, short and strong inputs from CA3. Different 753 CA1 PCs receive these inputs at different times, their density is highest near the peak 754 of the sharp wave. We propose that such sparse inputs might originate from dendritic 755 spikes in the apical dendrites, which are elicited when enough spikes arrive from CA3 756 within a short time window [65,67].

757
The inclusion of pyramidal cells and the condition of their sparse firing renders the 758 generation of high frequency oscillations challenging compared to models where the 759 oscillation is basically generated by I neurons only. This is on the one hand because the 760 additional E-to-I and I-to-E loop generally slows down the network oscillation 761 frequency [37]. On the other hand, we find that simply increasing the level of external 762 excitation to increase the oscillation frequency usually results in too much E cell spiking, 763 which renders the models unsuitable to describe ripple oscillations. Previous 764 two-population models for HFOs have assumed connection probabilities [27,29,37,117] 765 and/or spiking dynamics [25,29,37,117] that do not fit the situation in CA1 [1,20,57]. 766 Characteristic for the spiking activity during CA1 SPW/Rs is that they are a mixture 767 of two often considered oscillation types: strongly synchronized [36] and weakly 768 synchronized oscillations [25,27,37]. The E cell population is weakly synchronized; the 769 average single-cell firing frequency during a ripple is much lower than the population 770 frequency. The I cell population (consisting of PV+BCs) is strongly synchronized, with 771 every I cell spiking at nearly each individual ripple wave. In the current article, we 772 aimed at taking into account both the experimental knowledge on CA1 connectivity and 773 the characteristics of the E and I cell spiking activity during SPW/Rs.

774
How realistic are our assumptions concerning the generation of dendritic spikes?

775
Apical dendritic spikes have been directly observed during SPW/Rs [86]. It is difficult 776 to estimate the number of synaptic inputs required to generate dendritic spikes, 777 estimates range "anywhere from a handful to dozens of inputs" [118]. Detailed 778 multicompartmental models of morphologically reconstructed neurons suggested that at 779 least ∼ 50 synaptic inputs arriving within 1 − 3 ms on a small part of the apical 780 dendrite of CA1 neurons are required to elicit dendritic sodium spikes [65,67]. In our 781 model 3, we assume that already 5 spikes arriving within 2 ms are sufficient for 782 dendritic spike generation. This is similar to the number of inputs required for spike 783 generation in basal dendrites of CA1 pyramidal cells [43,44,66]. Increasing the 784 threshold for dendritic spike generation while keeping the number of afferent inputs and 785 the dendritic integration window w D constant necessitates an increase of the afferent an increase of r 0 in Eq 7 from 8 Hz to more than 100 Hz to approximately maintain the 788 same number of dendritic spikes. This value is clearly too high as a discharge rate for a 789 typical CA3 cell during sharp waves [64] (but values larger than 10 Hz are possible [60]). 790 The inputs from CA3 to CA1 may, however, be clustered such that sufficient coincident 791 inputs impinge on an apical dendrite. Our result might also indicate that a smaller 792 number of coincident inputs than previously estimated is required to elicit dendritic 793 spikes. Further, dendritic spikes relevant during SPW/Rs may be less sensitive to 794 synchrony than assumed in our model, i.e. their effective dendritic integration window 795 may be longer. Such dendritic spikes might then be NMDA or calcium 796 spikes [71,86,[119][120][121]. CA3 cell bursting [60] might in principle also contribute to 797 dendritic spike generation, for example because asynchronous but overlapping bursts 798 generate synchronous spike inputs. Experimentally found ISIs within individual CA3 799 bursts [60] are, however, typically larger than the sodium dendritic spike integration 800 window [65,66]. Finally, basal dendrites could generate dendritic spikes as incorporated 801 in our model, since they receive inputs from CA3 PCs besides that from CA1 PCs [15]. 802 Particularly promising candidates are the recently discovered axon carrying dendrites, 803 from which the axon emanates in many CA1 PCs [69]. These dendrites are particularly 804 excitable, generate strong dendritic spikes and have a high impact on action potential 805 generation. Ref. [22] found that stimulation of the local E CA1 population can excite 806 ripple oscillations. Such stimulation may generate dendritic spikes in the basal dendrites 807 of postsynaptic CA1 E cells [43,44], which could lead to ripples in a similar manner as 808 in our model 3 (or in the manner described in ref. [43,44]).

809
PC spiking during SPW/Rs in our model 3 is generally very sparse, the majority of 810 PCs contributes one to two spikes, as observed experimentally. In the model without 811 sequence replay a few CA1 PCs spike more than 3 times during a SPW/R event (Fig 7). 812 Such bursting is consistent with experiments [39,60,100,101]. The bursts are generated 813 in our model because the lognormal distribution of the peak dendritic currents across 814 neurons (Eq 2) contains larger values with non-negligible probability. Additional 815 simulations with truncated lognormal or Gaussian distributions with less bursting and 816 simulations with a long somatic refractory period, which completely prevents multiple 817 spiking, can generate ripple range HFOs. This shows that bursts are not necessary for 818 the generation of HFOs in the ripple frequency range in our model.

819
HFOs in the ripple range occur in model 3 (Fig 8), at or before the border of

827
A prominent property of the hippocampus is that it generates sequences of activity. 828 These may replay previously imprinted sequential experience [101,103,104], serve as a 829 backbone to store episodic memories [122,123] or generally provide a sequential 830 reference frame for sensory experience and brain activity [124]. Sequence generation is 831 often assigned to CA3 because of its prominent, albeit sparse, recurrent excitatory 832 connectivity [122,123]. However, CA1 also generates sequences on its own [39]. Earlier 833 work proposed that the highly sparse recurrent excitatory connectivity in CA1 could 834 underlie sequence generation, since it may be highly structured and amplified by basal 835 dendritic spikes [43,44]. Here we propose a different class of models for sequence 836 generation in CA1: two-population models. In such models the sequences are generated 837 cooperatively by the E and I population. We conceptually propose two such models, one 838 where the sequence generation depends on the prominent E-to-I neuron and I-to-E 839 June 4, 2021 29/55 neuron connectivity and another one where also the similarly prominent I-to-I neuron 840 connectivity is important. We explicitly implemented the first one of these concepts, 841 since it is more plausible for CA1. It is based on alternating pulse and gap coding of the 842 E and I neuron populations, such that both the E and the I neuron firing patterns 843 together generate the sequence. This is different from the classical view that sequence 844 generation in neural networks depends mainly on the excitatory connectivity between E 845 neuron groups like in synfire chains [125][126][127][128], while inhibition prevents pathological 846 activity, allows gating and introduces competition between sequences [129][130][131][132][133][134][135].

847
Specifically, inhibition of the embedding network and of previously active groups by 848 propagating synfire chain activity prevents pathological, strong increases of overall 849 network spiking activity [129,133]; the stability of propagation along the synfire chain 850 can then be improved by additional inhibitory neuron groups: their sparse feedforward 851 activation leads to disinhibitory removal of excessive inhibition from specific excitatory 852 groups [135]. We have shown that two-population based sequential replay is compatible 853 with filtering of CA3 input by dendritic spikes, as proposed in model 3, and that it 854 generates high frequency oscillations with sparse E cell firing. We expect that the latter 855 holds also for different forms of input from CA3 such as those explored in model 1.

856
Two-population based sequential structures may be present in further areas beyond 857 the hippocampal area CA1. They may consist of discrete groups or be continuous and 858 they may be preexisting or learned during experience. It is an interesting direction of 859 future research to determine how their spontaneous formation or learning may take 860 place through the interplay of excitatory and inhibitory synaptic plasticity [44,[136][137][138]. 861 To conclude, based on neurobiological knowledge on the hippocampal regions CA1  Table 1. Parameters of E cells. Table 2. Parameters of I cells.     Fig 16). Parameters as in Fig 4, except for drive of I cells: Each PV+BC is driven by a conductance as in Eq 6, with width σ g = 10 ms and fixed amplitudeḡ = 20 nS. In Figs 17 -20, the frequency range for f I and f E is set to [100,200]  Higher p IE , truncation of distribution for I peak dendritic , higher I-to-E synaptic     HFOs in networks incorporating dendritic excitation and a Gaussian distribution for the peak dendritic current as well as lower E-to-I peak conductance. The white circle is located at (σ, µ) = (0.75, 0.0) nA. Parameters as in Fig 8, except for a Gaussian distribution of the peak dendritic current (I peak dendritic ∼ N (µ Gaussian (µ, σ), σ Gaussian (µ, σ)) and lower E-to-I peak conductance g I exc,peak = 1 nS instead of 3 nS. The mean and standard deviation of the Gaussian distribution are matched to the mean and standard deviation of a lognormal distribution with parameters µ and σ: µ Gaussian (µ, σ) = exp µ + σ 2 2 , σ Gaussian (µ, σ) = exp σ 2 − 1 exp 2µ + σ 2 ; to allow a direct comparison with the previous figures, for parameters µ and σ are used for the plot axes.   Each group E k projects to one group of inhibitory neurons, to I k , as displayed for E 1 . I k projects to all E l with l = k + 1 as displayed for I 1 . The green dashed line denotes the absent connections from group I 1 to group E 2 . Group E 0 is stimulated to fire first and the replay event progresses along the black arrow: E 0 activates I 0 , which inhibits all E cells except those of E 1 . Thus, in the next step the group E 1 becomes active and stimulates I 1 , which leads to activation of E 2 etc. B: Scheme for sequence generation by gap coding of I cells. In contrast to Fig 9 A and panel A of this figure, the sequence generation relies on the inhibitory population alone, no E-to-I connections are necessary. The kth replay step is in the inhibitory neurons reflected by a deactivation of group I k , as in Fig 9 A. There are K + 1 inhibitory groups I 0 to I K . Neurons of group I 0 project to all inhibitory neurons except to those of I 1 (displayed by dashed green line). Similarly, group I 1 does not project to group I 2 , but to all other inhibitory groups. Group I 1 thus receives projections from groups I 1 to I K , group I 2 receives projections from groups I 0 and I 2 to I K and so on. Every inhibitory group gets projections from K inhibitory groups (in the figure: K = 4). In the first replay step, group I 0 is silent, whereas all other groups fire. In the next step, every group I l , l > 0, except I 1 receives the same amount of inhibition from only K − 1 groups, namely from the previously active groups I 1 to I l−2 and I l to I K , which project to it. The input from I 0 is missing, since it was not active. I 1 receives more inhibition than the other groups because all its K inhibitory presynaptic groups were active in the previous step (the silence of I 0 does not reduce I 1 's inhibitory input, since I 0 does not project to I 1 ). Thus, after the zeroth cycle, group I 1 is silent while the other groups are active. In the next cycle, group I 2 will be silent because it received more inhibition than the other groups and so on. We note that I 0 also gets inhibitory input from K groups in each step because it has no predecessor group not inhibiting I 0 but all other groups; its neurons should thus receive a compensatory excitatory input throughout sequence generation to be active after the zeroth step. The E cells in group E k can be entrained in this scheme by disinhibition, via inhibitory connections from group I k to E k .