How activity dependent feedback inhibition may maintain head direction signals in mouse presubiculum

Orientation in space is represented in specialized brain circuits. Persistent head direction signals are transmitted from anterior thalamus to the presubiculum, but the identity of the presubicular target neurons, their connectivity and function in local microcircuits are unknown. Here we examine how thalamic afferents recruit presubicular principal neurons and Martinotti interneurons and the ensuing synaptic interactions between these cells. Pyramidal neuron activation of Martinotti cells in superficial layers is strongly facilitating such that high frequency head directional stimulation efficiently unmutes synaptic excitation. Martinotti cell feedback plays a dual role: precisely timed spikes may not inhibit the firing of in-tune head direction cells, while exerting lateral inhibition. Autonomous attractor dynamics emerge from a modeled network implementing wiring motifs and timing sensitive synaptic interactions in the pyramidal - Martinotti cell feedback loop. This inhibitory microcircuit is therefore tuned to refine and maintain head direction information in the presubiculum.


INTRODUCTION
The neural head direction signal is processed over several interconnected brain areas, and similarly to other sensory systems, it is relayed through the thalamus 1 . From there it reaches the presubicular cortex, located between the hippocampus and the entorhinal cortex 2 . About half of presubicular principal neurons signal head direction 3,4 . They fire persistently when the head of the animal faces a specific direction. The dorsal presubiculum, also termed postsubiculum (Brodmann area 48), controls the accuracy of the head direction signal and links them to specific features of the environment thus enabling a role for the hippocampal formation in landmark-based navigation 5,6 . Vestibular inputs make a decisive contribution to head directional firing of neurons in the anterodorsal nucleus of the thalamus 1,7,8 and lesions of this thalamic region abolish head direction firing in presubiculum 6 . Head direction signals transmitted via the thalamus are integrated in the presubiculum with visual information 5 from visual 9 and retrosplenial cortices 7 , and information from the hippocampal formation 2 . Presubicular head direction cells in layer 3 project to the entorhinal cortex 10,11 and may contribute to spatial firing of grid cells [12][13][14] .
The properties of presubicular microcircuits that signal head direction are less clear than the longrange outputs from the region. The electrophysiological and morphological properties of excitatory and inhibitory presubicular neurons have been described 15,16 . Pyramidal cells can generate persistent firing with little adaptation over tens of seconds 17 as needed to signal a maintained head direction.
However, less is known of the connectivity and dynamics of inter-and intralaminar presubicular synapses 18 . Such data are crucial to understand how signals are transformed within the presubiculum and how this structure gates the flow of head direction information to the entorhinal cortex.
The roles of presubicular interneurons are presumably multiple: they provide global inhibition to restrain over-excitation 19 and, as suggested by continuous attractor theories, could induce selective inhibition of pyramidal cells, ensuring head direction signal specificity over time 14,[20][21][22][23] . Yet, details of the recruitment of inhibitory cells are unknown. In somatosensory cortex, high frequency pyramidal cell firing is needed to recruit Martinotti interneurons. These cells then initiate a feedback inhibition of distal pyramidal cell dendrites [24][25][26] , to exert a local control on excitatory synapses made at these sites 27 .
Facilitating excitation of interneurons may be critical for the treatment of the persistent head direction signal, however, there is no data on the functional effects of Martinotti cells in the presubiculum.
We report here that strong recurrent connectivity between the presubicular Martinotti cells and layer III pyramidal cells form a feedback inhibitory circuit. Importantly, the excitation of Martinotti cells by pyramidal cells exhibits a dramatic activity-dependent facilitation. The feedback effects of Martinotti cell inhibition on pyramidal cell activity depend on IPSP timing, suggesting they could provide a source of lateral inhibition that enforces directionally selective firing. Testing these hypotheses by modelling connectivity and synaptic dynamics of recurrent Martinotti-cell mediated inhibition revealed features of an attractor network generating activity patterns comparable to presubicular recordings in vivo. Our results demonstrate autonomous dynamic activity in the presubicular cortex emerging from the local circuits that process head direction signals in vivo.

Electrophysiology of presubicular Martinotti and pyramidal cells
In order to elucidate the functional role of Martinotti cells in the presubicular microcircuit, we first characterized the basic properties of Martinotti cells (MC) and pyramidal cells (PC) in superficial layer 3 of mouse presubiculum. In total, data from 166 PCs and 161 MC recorded in horizontal slices ( Fig. 1) from 60 animals are included in this study. MCs were identified as GFP positive neurons in tissue from X98-SST and Sst-Cre::tdTomato transgenic mice 16 . Martinotti cells often discharged spontaneously from a relatively depolarized membrane potential above -60 mV (n = 80 cells; Fig. 1a 16 ). They exhibited low threshold spiking in response to current pulses and their axons ramified extensively in layer 1 like Martinotti cells in somatosensory cortex 28 (Fig.   1a,d,e). Pyramidal cells, in contrast, typically did not discharge spontaneously and membrane potentials were significantly more hyperpolarized, below -70 mV (n = 87 cells), than those of MCs (Mann Whitney test, two-tailed p < 0.0001; Fig. 1a-c and Supplementary Table 1). PCs fired regularly in response to injected current 15 with a higher threshold current (92.3 ± 50.4 pA, mean ± sd; n = 65) than in MCs (51.5 ± 38.9 pA, mean ± sd; n = 64; Mann Whitney test, two-tailed, p < 0.0001). The input-output gain was lower in PCs (0.373 ± 0.127 Hz.pA -1 , mean ± sd; n = 65) than in MCs (0.845 ± 0.040 Hz.pA -1 , mean ± sd; n = 64; Mann Whitney test, two-tailed, p < 0.0001, Fig. 1e and Supplementary Table 1).

Anterior thalamic fibers directly excite principal neurons in superficial layers of presubiculum
Head directional inputs to the presubiculum originate in part from the Anterior Thalamic Nuclei 6,29 (ATN). We sought to define presubicular targets of these afferents by in vivo stereotaxic, intrathalamic injection of viral vectors to transduce channelrhodopsin-2 fused to eYFP ( Martinotti-like neurons optically evoked EPSCs occurred with longer, more variable latencies (3.5 ± 0.7 ms) with lower charge transfer (median = 0.200 nC over 25 ms) indicating a weaker excitatory drive (Fig. 2c,e,f). TTX (1 µM) and 4AP (100 µM) let us examine synaptic excitation mediated by thalamic afferents in isolation. Optical stimulation continued to excite PCs (median = 1.306 nC over 25 ms, Fig. 2d,g) showing they are directly innervated by ATN fibers, but light-evoked responses in MCs were suppressed (median = 0.013 nC over 25 ms, Fig. 2d,g). These data suggest that optical excitation of Martinotti cells is mediated indirectly via synapses made by presubicular pyramidal cells.
In return, Martinotti cells may provide a recurrent inhibitory control of pyramidal cells. We next examined the presubicular PC-MC connectivity pattern and its dynamics using dual patch clamp recordings in X98-SST mice for Martinotti cell identification.

A feedback loop: pyramidal cells activate Martinotti cells and are inhibited in return
Pyramidal cells and Martinotti cells were highly interconnected (Fig. 3) as expected from the spatial overlap of their axons and dendrites (Fig. 1a). The proportion of connected pairs was 57% (83 of 146 tested) for Martinotti cell to pyramidal cell (MC-to-PC) and 38% (59 of 156 tested) for pyramidal cell to Martinotti cell (PC-to-MC). 28% of cell pairs (39 of 141) were reciprocally connected, a little more than the 22% expected given the probability for unilateral connections. For PCs that excited a MC, the probability of reciprocal inhibitory connection was very high, 81% (39 out of 48 tested), while only 48% (39 out 80 tested) of MCs inhibiting a PC received reciprocal excitation. Connectivity between pyramidal neurons (PC-to-PC) was very low (1 of 48 tested). At -50 mV, the mean amplitude of inhibitory postsynaptic currents or potentials (IPSCs or IPSPs) triggered by Martinotti cells was 9.01 ± 1.19 pA (n = 45) or -0.56 ± 0.07 mV (n = 21). The probability that a single spike triggered a postsynaptic event was high (transfer rate 0.86 ± 0.05, mean ± sem; median = 0.925; n = 11; Fig.   3c,d), and for multiple trials, at least one postsynaptic event was observed for each connected pair. PCto-MC transmission was much less reliable. For single spikes, the transfer rate from pyramidal cells to Martinotti cells was very low, 0.12 ± 0.02 (median = 0.08; n = 44, Fig. 3e). In 6 pairs, successful synaptic transmission occurred only during high frequency trains, which allowed us to identify them as functionally connected pairs, but single presynaptic spikes never initiated a postsynaptic response (at least 30 trials for each pair). In 38 PC-to-MC pairs, single pyramidal cell spikes, or first spikes in a train, occasionally initiated excitatory postsynaptic responses. Their potency, that is, the mean absolute amplitude of single successful responses for PC-to-MC synapses, was 20.1 ± 1.94 pA (median = 20.4 pA; n = 31) or 1.44 ± 0.21 mV (median = 1.37 mV; n = 8). The efficacy, the potency multiplied by the transfer rate, was 2.36 ± 0.58 pA (median = 1.24; n = 38) or 0.24 ± 0.05 mV (median = 0.27; n = 9) for the first spike; Fig. 3e; Supplementary Fig. 2).
We have shown a significant asymmetry in synaptic reliability in the recurrent inhibitory loop between pyramidal cells and Martinotti interneurons in superficial layers of the presubiculum: inhibitory synapses are much more reliable than excitatory connections. Since the dynamic behavior of both synapses in this feedback circuit will govern its operation 30,31 , we examined postsynaptic responses at different rates of pre-synaptic firing. Transfer rate, potency and efficacy were analysed for synaptic responses to trains of 30 action potentials at either 10 Hz or 30 Hz, repeated with an inter-stimulus interval of at least 20 seconds ( Fig. 4a and 5a). We detected all postsynaptic events and classed those following pre-synaptic spikes at mono-synaptic latencies (see methods) as spike-induced events.

Stable Martinotti cell inhibition during repetitive stimulation.
Information transfer at MC-to-PC connections was reliable and stable during synaptic activation at 10 or 30 Hz (Supplementary Table 2 and Fig. 4a-d). The synaptic efficacy for the first five action potentials (early efficacy) and that of the last five action potentials (late efficacy) in trains of 30  pairs; Friedman test, P = 0.5222). Changes in efficacy during repetitive firing were mostly due to changes of potency and less to alterations in transfer rate (Fig. 4e). Cumulative efficacy evolved linearly during repetitive stimulations (Fig. 4f). Changes in synaptic frequency (see methods) were proportional to changes in presynaptic firing frequency (Fig. 4g). Thus, the dynamic behavior of MCto-PC inhibitory synapses is relatively stable with little dependence on the history of pre-synaptic firing.

Repetitive stimulation unmutes the PC-to-MC connection in a frequency dependent manner
In contrast, PC-to-MC excitatory synapses displayed remarkable facilitating dynamic behavior (n = 58/59 pairs). Figure 5 shows an example of excitatory postsynaptic currents (EPSCs) elicited by 10 Hz and 30 Hz stimulations (Fig. 5a). Synaptic efficacy was low at first, but increased greatly with both the number and frequency of pre-synaptic action potentials, even though spike-to-spike responses varied between trials ( Fig. 5a-d, Friedman test, P = 0.0002). At 10 Hz, late efficacy (6.51 ± 2.99 pA, mean ± sem, n = 9) was more than double early efficacy (2.99 ± 0.99 pA, mean ± sem, n = 9). At 30 Hz, late efficacy (17.34 ± 5.14 pA, mean ± sem, n = 9) was four times higher than early efficacy (4.41 ± 0.88 pA, mean ± sem, n = 9). Efficacy increased for 6/9 pairs tested at 10 Hz (Fig. 5d left, Dunn's multiple comparison, n.s.) and for 9/9 pairs at 30 Hz ( Fig. 5d right, Dunn's multiple comparison, p < 0.05). The extent of the increase varied between connections especially at 30 Hz (Fig. 5d). In contrast to the MCto-PC synapse, changes in efficacy at the PC-to-MC synapse during activation at 30 Hz were due to variations in response probability and not in potency (Fig. 5e, n = 15 pairs). Increased efficacy implies a greater reliability of PC-to-MC synaptic transmission for increasing numbers and frequencies of presynaptic spikes. Furthermore the synaptic frequency increased supra-linearly with presynaptic spike frequency. After one second, the cumulative efficacy was 11 times higher at 30 Hz than at 10 Hz (Fig.   5f). The synaptic frequency was 5.9 times faster for early spikes and 7.9 times faster for late spikes, when presynaptic firing rate increased from 10 to 30 Hz (Fig. 5f,g), thus largely exceeding expected changes due to a three-fold increase in the rate of synaptic activation. We refer to this remarkable form of facilitation as synaptic unmuting.

Increase of transfer rate at the PC-to-MC synapse as a medium term memory process
Presubicular Martinotti cells are reliably excited only when pyramidal cells fire at high frequency, as when they signal a preferred head direction. These frequency dependent changes do not reflect longterm synaptic plasticity since synaptic efficacy returned to previous values within ~20 s (Fig. 5a).
Several paired records nevertheless revealed medium term effects on the PC-to-MC excitatory synaptic transmission (n = 10, Fig. 6 and Supplementary Fig. 3). When synaptic unmuting was induced after initial 30-40 Hz high frequency firing of the pyramidal neuron, synaptic transfer at the PC-to-MC synapse remained enhanced even as PC firing adapted to lower frequencies of firing We used spike trains from isolated single head direction units within their preferred range in vivo as depolarizing current commands to presynaptic PCs in paired PC-MC recordings (Fig. 7c,d). As expected, excitatory transmission induced at the start of high frequency in vivo spike patterns was poor. The PC-to-MC synaptic efficacy increased considerably during sustained high frequency firing (from 0.9 ± 0.6 to 21.7 ± 11.9 pA; n = 5), and synaptic unmuting persisted during later sparse firing, even after a silent period of several hundred milliseconds (Fig. 7e,f). We noted not only an increase in synaptic events "locked" to presynaptic spikes with latencies < 3 ms, but also an increase in the frequency of delayed excitatory postsynaptic events after sustained high frequency firing (Fig. 7e).
Firing of Martinotti cells induced by pyramidal cell firing was consistent with facilitating synaptic dynamics. Synchronous and asynchronous EPSPs summed to reach Martinotti cell firing threshold during repetitive high frequency firing (Fig. 7g,h).
For comparison, we also examined the synaptic transmission of the same spike train onto fast-spiking parvalbumin (PV) expressing interneurons in paired PC-PV recordings ( Supplementary Fig. 4). Quite opposite to Martinotti cells, PV+ neurons responded with highest efficacy at the onset of a high frequency spike train, then displayed depression. The facilitating pattern of synaptic recruitment was therefore specific to Martinotti cells but not PV+ interneurons.

Spike timing dependent inhibitory effect favoring lateral inhibition over self-induced inhibition
We next asked how Martinotti cell mediated feedback IPSPs affected post-synaptic pyramidal cells.
Pyramidal cell spikes typically initiated Martinotti cell firing at a less than 8 ms delay (84% of spikes, n = 4 pairs; Fig. 8a, right panel). Therefore, in reciprocally connected cell pairs the great majority of self-induced Martinotti-cell mediated IPSPs occurred with short latency (<10 ms) after a PC spike.
These feedback IPSPs coincided with the spike afterhyperpolarization (AHP) of the triggering pyramidal cell. As illustrated in Fig. 8a, feedback IPSPs summed with the AHP, resulting in a larger pyramidal cell hyperpolarization and enhancing the peak amplitude of the next pyramidal cell action potential, while exerted little inhibitory effect on pyramidal cell firing.
However, Martinotti cells also mediate lateral inhibition. IPSPs in neighboring, but not reciprocally connected, pyramidal cells tend to occur with timing unrelated to preceding pyramidal cell spikes. The functional difference between feedback inhibition versus lateral inhibition on PC firing can therefore be addressed by studying the effect of short (<10ms) versus long latency IPSPs. We tested the hypothesis that lateral inhibition has distinct effects to reciprocal inhibition in recordings from unidirectionally connected MC-to-PC pairs with mean IPSP amplitude greater than -0.3 mV at -50 mV (n = 7; Fig. 8b-e). MC action potentials were timed to initiate IPSPs at different times during the PC firing cycle (n = 7, 30-50 Hz). We then compared the effects of IPSPs of latencies <10 ms (feedbacklike) or >10 ms between PC and MC firing. We measured values for the pyramidal cell AHP (AHP TEST ) together with the peak of the next spike (PK TEST ) and the inter-spike interval (ISI TEST ).
Short latency feedback IPSPs are induced when persistent PC firing recruits a MC -these IPSPs have little inhibitory effect and may even encourage PC firing. In contrast, delayed IPSPs impinging on non-reciprocally connected PCs tend to delay subsequent PC discharges.

Inhibitory attractor network model reproduces presubicular head direction signaling
Head direction signals are organized internally, such that neurons with similar preferred head directions fire together in a correlated way 29 . Computational models of the head direction signal suggest that this activity profile may emerge from an attractor network 20,23,32,33 . These models mostly rely on strong excitatory connections between cells with similar preferred directions. We asked We show that, in the absence of correlated inputs, the model network spontaneously generated a directionally selective increase in activity, thus satisfying attractor network dynamics (Fig. 9b). The model neurons coding for a certain direction forcedly mirrored the thalamic directional input, and when the external drive was reduced and the system relaxed, the neuronal activity profiles were mostly maintained (Fig. 9c). Polar plots of the activity of representative pyramidal cells were similar to those of finely tuned head direction cells in vivo, while Martinotti cells were very little directionally modulated ( Supplementary Fig. 5). The precision of the pyramidal cell tuning could be controlled by varying the range α of the inhibition suppression around an existing connection between a pyramidal cell and a Martinotti cell. Finally the model allowed us to test the importance of the facilitating synaptic dynamics of Martinotti cell recruitment for the formation of a coherent activity bump. When facilitating synapses were replaced with depressing or stable synapses, bump formation could be obtained if synapses rapidly returned to their initial state (large b1, Supplementary Fig. 5). However, the activity correlation with initial external input fell apart, underscoring the key importance of the facilitating PC-to-MC synaptic properties for a maintained head directional signal (Fig. 9d,e). Thus recurrent excitatory synapses made with PV interneurons, which exhibit a dynamic depression, are not part of the attractor that maintains the head directional information in the presubiculum. In conclusion, an inhibitory feedback triggered exclusively at high firing frequencies with spike-timing dependent inhibitory effects on pyramidal cells will suffice to refine and sustain head direction signals in the presubiculum.

DISCUSSION
We have described activity-dependent dynamic properties of the Martinotti cell inhibitory feedback loop in the presubiculum. These properties underlie a self-sustained processing of head direction information in presubicular microcircuits. Superficial pyramidal cells are directly excited by thalamic inputs. Martinotti type interneurons are excited by these pyramidal cells and reliably inhibit pyramidal cell dendrites in layers 1 and 3. Feedback excitatory transmission from pyramidal cells to Martinotti cells is greatly facilitated during sustained high frequency presynaptic firing. Synaptic transfer may be enhanced for several seconds after a PC-to-MC connection is "unmuted". The behavior of this feedback inhibitory circuit is directly relevant to patterns of head direction activity. Natural firing patterns of these cells, recorded in vivo, recruited Martinotti cells very effectively in vitro whereas lower firing frequencies had little effect. Firing of these interneurons had distinct timing-dependent effects. In reciprocal connections, MCs fired at short latencies after PC action potentials. Inhibition by such precisely timed, spike-locked IPSPs was less effective than for randomly timed IPSPs, such that Martinotti cells provide a strong lateral inhibition. This feedback circuit is well-adapted to refine head direction signals in the presubiculum and to robustly preserve sustained firing of in-tune head direction cells.
Head direction signals are thought to be generated in subcortical nuclei and relayed via the thalamus to the parahippocampal region 1,6 . Neurons of anterior thalamus (ATN) project quite specifically to the presubiculum 7 (Fig. 2). A monosynaptic connection from ATN to presubicular head direction cells has been recently inferred in vivo based on short latency, reliable spike transmission 29 . Here we examined the effects of optogenetic activation of anterior thalamic axon terminals on single presubicular neurons in vitro. Our data provide functional evidence for a direct innervation of layer 3 pyramidal neurons of the presubiculum by thalamic fibers. Martinotti type interneurons received no direct excitation.
Pyramidal cells of superficial layers project directly to the MEC (data not shown; cf. also 11,34 ). While grid cell activity of MEC neurons depends on head direction information 35 , the ATN does not project directly to the MEC. Thus integration of head direction code in presubicular superficial layers seems to be an essential element in the construction of inputs to MEC grid cells.
Recurrent feedback circuits of Martinotti cells and pyramidal cells are highly interconnected. The probability of PC-to-MC connections was 37%. The MC-to-PC connection probability was even higher: 58%. Such estimates from paired recordings are probably underestimates since all connections may not be preserved in slices. Our pipette solution was designed to enhance the driving force for chloride, increasing our ability to detect inhibitory synaptic events and to distinguish them from failures. Nevertheless, we may have missed low amplitude inhibitory synaptic events generated at very distal dendritic sites. Martinotti cells of other cortical areas also have high connection probabilities with local pyramidal cells to provide a dense, reliable and non-specific inhibition 36 , with both convergent and divergent connectivity 25,37 . We detected no direct activation of Martinotti cells by thalamic afferents reinforcing the feedback role of MCs in a presubiculum circuit. With a very low rate of recurrent connection between pyramidal cells (~ 2%), the PC-MC pathway becomes especially important to mediate interactions between presubicular pyramidal cells, similar to layer 5 pyramidal cells in neocortex 25 or to layer 2 stellate cells in medial entorhinal cortex 38,39 .
We found MCs were only excited to fire by summed EPSPs induced after synaptic unmuting when PCs fired at high frequencies for prolonged periods. Single PC spikes never led to MC discharge (Fig.   7g, 8a). The short-term dynamics of pyramidal cell synapses vary between fast-spiking, parvalbumin expressing or low threshold spiking, somatostatin expressing interneurons in neocortex and hippocampus 25,40-42 (cf. also Supplementary Fig. 4). The facilitation during repeated activation shown here at synapses that excite Martinotti cells is similar to that of synapses made with SST immunopositive interneurons in hippocampus 41 and neocortex 24,25,42,43 , even though the presubiculum is not a typical neocortical area, but rather part of the transitional periarchicortex. Synaptic facilitation in the presubiculum has slow kinetics, corresponding well to the persistent discharges of head direction cells. In somatosensory cortex, a 3-fold increase in average EPSP amplitude could be obtained after 8 stimulations at 20 Hz 43 , while in presubiculum a similar degree of facilitation was obtained after 30 stimuli at 30 Hz. Presubicular PC-to-MC synapses were often silent during paired pulse stimuli. We therefore analyzed synaptic dynamics from responses to trains of action potentials at 10 or 30 Hz. Enhanced synaptic efficacy during these trains resulted from increased transfer rate rather than potency (Fig. 5e). This phenomenon persisted for a time after high frequency stimuli ( Fig. 6 and Supplementary Fig. 3) as at some other synapses 31,44 . Nevertheless even after unmuting, the transfer rate remained quite low at this synapse, compared to responses elicited by similar stimuli at neocortical PC-to-MC synapses in layer 3 43 or layer 5 25 .
Possibly presubicular PC-to-MC transmission is regulated by an activity dependent mechanism, distinct from short term facilitation 31,45 , situated at either axonal or presynaptic sites 46  Experimental data let us propose a modified continuous attractor model based on recurrent inhibition to mimic head direction activity. The build-up of strong principal neuron activation as a necessary condition for interneuron recruitment is essential to the model. We suggest that activity-dependent unmuting of Martinotti cells and their facilitating synapse dynamics are key for autonomous circuit dynamics in the presubiculum. In contrast to primary visual cortex 54 , the presubiculum could, in this way, sustain activity. Fast-spiking PV neurons with depressing synapses (Supplementary Fig. 4) are not part of the attractor. We suggest that during fast head turns, when Martinotti cells are not recruited, the system may switch to a relay type function. Presubicular fast-spiking interneurons fire at higher rates during rotation 10 , that is, when the population of active head direction cells shifts quickly. The excitatory inputs received by PV neurons continuously changes to different sets of synapses, and for each transient head direction, PV neurons will rapidly provide inhibition with depressing dynamics.
However, Martinotti cells become active and stay active during maintained directional signaling, and could support a form of working memory [55][56][57] , especially in the absence of a stabilizing stimulus 55 . Our data on a one-dimensional head direction system might suggest that equivalent dynamics exist in the medial entorhinal grid-cell system 38,58 . We note the model network requires no directional tuning of presubicular interneurons. The efficacy of inhibitory synapses depends exclusively on the timing of interneuron firing with respect to firing in the presynaptic principal neuron, leading to little inhibition for a driving head direction cell, but stronger lateral inhibition. In conclusion, the recruitment of Martinotti cells by differentially active, randomly connected pyramidal cells provides an economic way to refine and sustain presubicular head direction signal representations.

Stereotactic Virus Injections
Adeno-associated viral vectors carrying genes for ChR2-EYFP fusion proteins (AAV2/9.hSyn.hChR2 (H134R)-EYFP.WPRE.hGH; University of Pennslvania Vector Core) were injected into the anterior thalamic nucleus (ATN) at postnatal age P28. For surgery, mice were deeply anesthetized with intraperitoneal injection of ketamine hydrochloride and xylazine (100 and 15 mg.kg -1 , respectively) following stereotaxic procedures described previously 62 . The virus was delivered via a 33-gauge needle with a Hamilton syringe in a syringe Pump Controller (Harvard Apparatus, Pump 11 elite) at 20 nl.min -1 . ATN was targeted at coordinates from Bregma: lateral, 0.75 mm; posterior, -0.82 mm; depth, -3.2 mm. Slices were prepared at 12-16 days after vector injection. The injected volume was 150 nl, to be as specific as possible (cf. Supplementary Fig. 1), but with enough spread to cover ATN. Martinotti-like cells (MC) of tissue from X98-SST mice were defined as green fluorescent neurons, and those from Sst-Cre::tdTomato mice as red fluorescent neurons. In both mouse lines, MC possessed resting membrane potential above -65 mV. Discharges were either adapting or low threshold firing and biocytin filling revealed typical Martinotti cell axonal and dendritic morphologies (Fig. 2).

Slice preparation, in vitro electrophysiology and photostimulation
Channelrhodopsin expressing terminals from the AT thalamic nucleus were excited with blue light from a source (Cairn OptoLED, white) coupled to the epifluorescence microscope port, filtered (BP 450-490, FT 510) and fed into a 60X 1.0 NA plan-Apochromat objective. Light pulses of 0.5 ms duration and intensity 2 mW were delivered at 20 s intervals. 1 µM TTX and 40 µM 4AP were added to the bath to check for direct vs. indirect optical activation. Salts and anesthetics were all obtained from Sigma, except TTX from Tocris.

In vivo electrophysiology
Head direction firing was sampled from presubicular neurons in vivo in rats, with higher channel counts and unit yield compared to mice (cf. ref. 3 ). Briefly, tetrodes were implanted in 4 months old Long-Evans rats at AP 2.2 mm in front of the transverse sinus, ML 3.7 mm from the midline, and DV 1.5 mm below the dura. Tetrodes were lowered progressively until reaching presubicular layers.
Recording sites in presubiculum were confirmed from post-hoc Nissl, parvalbumin and calbindin stained sections. Head direction was tracked with two light-emitting diodes while the animal collected randomly distributed food crumbs from a 100 cm wide square box. Spikes were sorted offline with cluster cutting Axona software. Head direction was calculated from projections of the relative position of the two LEDs on the horizontal plane. Directional tuning for each cell was obtained by plotting firing rate against the rat head direction, divided into bins of 3 degrees and smoothed with a 14.5 degrees mean window filter (14 bins on each side). Command protocols for slice records were generated from these spike trains imported into pClamp.

Data analysis
Signals were analyzed with AxoGraphX, and locally-written software (Labview, National Instruments; MATLAB, The Mathwork). Algorithms to detect action potentials and measure active and passive neuronal properties were described previously 15,16 . All relevant data are available from the authors.

Detection of postsynaptic events
Excitatory and inhibitory postsynaptic currents and potentials were detected and measured automatically from low-pass filtered records adapted to the recording mode (0.4 KHz for EPSPs, 1 KHz for EPSCs and 500-750 KHz for IPSCs). Spontaneous or spike-associated events were detected as continuous rising signals exceeding a threshold set for records from each cell to minimize both false positive and negative detection. Thresholds were 0.3-0.6 mV for EPSPs, 4-7 pA for EPSCs and IPCSs recorded in K-gluconate and 4-12 pA for IPSCs recorded with Cs-gluconate solution.
Spike-locked postsynaptic events were defined as first events occurring within a monosynaptic latency (generally 0.5-3 ms for an EPSC and 0.5-4 ms for an IPSC). Delayed postsynaptic events were those that occurred later than the spike-locked events or outside the monosynaptic window but still within 10 ms after the spike. PSC latencies were calculated from the action potential peak to the mid-rise of the postsynaptic event.
Spontaneous activity can bias values for synaptic transfer. We estimated "false positives" which might exaggerate monosynaptic transfer rates. Presynaptic firing patterns were aligned to a "control window", before stimulation, and transfer rate, corresponding to a noise value, was calculated. This procedure was applied multiple (250 -300) times using different starting points in the same control window. The number of "false positive" rarely exceeded 0.05. It depended on the level of background synaptic activity, but not on presynaptic firing frequency.

Synapse dynamics in repetitive stimuli
Synaptic transfer rate was calculated from paired records as the number of detected post-synaptic events divided by the number of presynaptic spikes. Failure rate was 1 -transfer rate. Synaptic potency (pA or mV) was defined as the amplitude of detected events. Efficacy (pA or mV) was the mean amplitude of responses including failures (failure amplitude = 0). Efficacy may be deduced as potency x transfer rate. Synaptic transmission during repeated presynaptic activation was analyzed in these terms to derive transfer rate, potency and efficacy for either (1) a given spike across different trials of a standard stimulus, or (2) groups of successive spikes elicited during a defined time. For spike trains, the first five spikes and the last five spikes were grouped to increase measurement precision (Figs. 4 and 5) when trial-to-trial variability was high. Changes of transfer rate, potency and efficacy over time are measured as the ratio of late/early values. Cumulative efficacy was calculated as the sum of efficacy over time and provides a temporal dynamic. The derivative of this cumulative efficacy, that we called "synaptic frequency", corresponds to the information transferred per second.

Inhibitory effect
Functional MC-mediated inhibition was quantified as the ability of an IPSP to delay PC-discharge (ISI modulation). We also measured effects of IPSPs induced after pyramidal cell firing. In an effective recurrent circuit, pyramidal cells may induce MC-spike firing evoking in turn an IPSP in the initiating pyramidal cell. This effect was quantified as an enhanced pyramidal cell AHP (AHP modulation) or change in peak amplitude of a PC-spike (Peak modulation). Both parameters could be affected by intrinsic properties such as adaptation, peak accommodation and an AHP depolarization during repetitive firing. We therefore determined the effect of inhibition as changes from predicted pyramidal cell repetitive firing behavior (Fig. 8).

Cellular anatomy
Biocytin (1mg/ml) was added to the pipette solution to reveal the morphology of some recorded cells as described 15,16 . Axo-dendritic morphology was reconstructed from z-stacks of acquired images with Neurolucida software (Microbrightfield, Williston, VT, USA).

Computational model
Our model builds on previous network models of the head direction system that generate attractor dynamics with directionality 21,63,64 . In the present model, network function is dominated by indirect, inhibitory interactions between pyramidal cells, and we explicitly include interneurons mediating such interactions in the dynamics of the system.
After establishing the inhibitory connectivity, the excitatory wiring was established as follows. Each main connection was associated with a reciprocal excitatory connection of uniform strength.
Additional excitatory connections were created with a 0.4 probability and a reduced strength, compared to those associated with the main connections, randomly drawn from the [0 w OUT ] interval (where w OUT < w ! OUT ) .
As a last step, the strength of the inhibitory connections converging on each pyramidal cell was normalized according to: (2) Ultimately, three different groups of connections could be found (cf. Fig. 9a): 1) strong main excitatory connections from pyramidal to Martinotti (purple arrow) 2) weaker excitatory connections (blue arrows) and 3) inhibitory connections (Network parameters are given in Supplementary Table 4).
Unit Dynamics. A pyramidal unit assigned with preferred direction was described by its firing rate at time t, , regulated through the following dynamics: where is a threshold linear f-I curve and is the neuronal time constant. The input to the unit consisted of an external input term, h (see below), and the contribution coming from feedback inhibition, I. In turn, the inhibition term consisted of the combined effect of the presynaptic Martinotti units: where is the strength of the inhibitory connection between Martinotti unit j and pyramidal unit . Similarly, for Martinotti units firing rate was regulated by the equation that includes a time constant generating slower input integration times. For the pyramidal units, the excitatory current E was the sum of pre-synaptic inputs. For Martinotti units, the summation was modulated by synaptic temporal dynamics described in the next paragraph. Synaptic facilitation may be crucial to stabilize the network [55][56][57] .
Dynamic properties of synapses. The formation of a coherent bump of activity crucially depended on the stable excitation of Martinotti interneurons. For Fig. 9d-e, in order to examine the influence of synapse dynamics on the stability of Martinotti interneurons activity, we modified the previous equation, (6) so that the term reflected either depressing or facilitating synaptic dynamics. We introduced a variable regulating the responsiveness of synaptic contacts between each pyramidal and Martinotti cell: The evolution of the synaptic efficacy was mediated by the equations: Act is a variable in the range [0,1] that integrates over time the synaptic activity between pyramidal cell Θ and Martinotti cell j . The activation state of the synapses is then turned into the efficacy value γ Θ, j Eff through a sigmoid transfer function. The two activation parameters control the direction (b2) and the persistence (b1) of the modulation. If b2 > 0 then synapses are facilitating, otherwise, when b2 < 0, the equations represent synaptic depression. For increasing b1, synapses come back faster to their initial state.
The experimentally observed spike-timing dependence of recurrent inhibition is implemented as the dynamic regulation of the effectiveness of feedback inhibition from a Martinotti unit to a pyramidal unit: Feedback inhibitory strength decreases as the contribution of a given pyramidal cell to Martinotti cell firing increases. The reciprocal feedback inhibition for each pair was re-computed at every timestep according to: Therefore, each pyramidal cell would see a modification of the inhibitory connection strength depending on its contribution to the activity of the corresponding Martinotti cell. In the case of a single pyramidal cell driving a Martinotti cell, its feedback inhibition would be zero.
External Input. Each pyramidal unit in the network received an independent, time-dependent, activation current from an external source. Since all the internal effective interactions between pyramidal units were inhibitory, this external source of excitation was necessary for activity in the network. In our simulations, each unit was fed with a random input, uncorrelated across units, but correlated in time: (12) where was a normal distributed random variable with mean and standard deviation . This random background input could be combined with an additional direction selective component, restricted to a sub-set of the units, centered around a given direction , where controlled the strength of this component (with respect to the background one) and regulated the degree of selectivity around the central selected direction (cf. Supplementary Table 4 for network parameters values).
Activity Bump Coherence. The degree of concentration of the pyramidal cell activity was measured as (14) Simulations. Simulations consisted in integrating the network dynamics during 200 time steps of 1-ms.
When only white noise was fed into the system, the simulation consisted in reproducing network dynamics starting from a random activity configuration. When studying the effect of directionally selective external inputs, the network received random and selective external inputs to the pyramidal cell layer for the first 60 time steps with the latter then gradually fading away between time steps 60 and 80. This procedure was repeated with the directional input sequentially centered over each of the cells preferred directions.
All simulations were performed using MATLAB custom code, available from the authors. Tables and five Supplementary Figures and can be found below.

Supplemental information includes three
h(Θ,t)       Magnification of a MC-spike evoked by PC firing at short latency (bottom). The MC-spike alters the PC-spike AHP (AHP TEST ), the PC-spike peak (PKTEST) and the PC ISI (ISI TEST ) according to the delay after the PC-spike PKINT. Dotted lines indicate the extrapolated level for PK TEST and for AHP TEST . Right, MC firing triggered by PC spikes in n = 4 PC-MC pairs. 84% of MC spikes have a delay of less than 8 ms after a PC spike (65 out of 77 spikes; n = 4 pairs).
(b) Spike timing dependent MC-inhibition was tested in unidirectionally connected MC-to-PC pairs. Drifting single MC-spikes were triggered during sustained PC firing . Two sweeps of PC firing are shown (one in blue, one in red), with the corresponding MC spikes at the bottom. For the blue voltage trace, the PC-spike to MC-spike delay was short ("time locked"; delay < 10 ms, similar to reciprocal connections as in (a)). For the red voltage trace ("delayed"), the MC spike delay exceeded 10 ms.
(c-e) Differential effect of short latency vs long latency inhibitory modulation of AHP, peak and ISI of PC spikes (30-50 Hz; n = 7). (c) The PC AHP was more hyperpolarized for short latency, time-locked MC-spikes but not for delayed MC-spikes (n = 7, * p < 0.05). The modulation of the PC-spike AHP was calculated as (AHP TEST -AHP INT )-(AHP INT -AHPREF). (d) The PC-spike peak after a MC-spike was higher for timed locked but not for delayed inhibition (n = 7, ** p < 0.01). Peak modulation was calculated as (PK TEST -PK INT )-(PK INT -PK REF ).
(e) The PC ISI increased more for delayed than for time-locked inhibition (n = 7, ** p < 0.01). ISI change was calculated as 100* (ISI TEST / ISI INT )/(ISI INT /ISI REF ). Each dot indicates the mean for one pair. Horizontal bars are medians. The median-null difference was assessed with a Wilcoxon signed rank test ( † p < 0.05) and the relative difference between short-and long-latency inhibition with a Wilcoxon match-pairs signed rank test (* p < 0.05, ** p < 0.01). ISI, inter-spike interval; AHP, after-hyperpolarization; PK, peak.