Cooperative molecular interactions are ubiquitous in biology and guide cellular processes from sensing to memory formation (Bray et al., 1998; Burrill and Silver, 2010). They are found not only in the simplest organisms like bacteria, but also in higher organisms including mammals (Stefan and Le Novère, 2013). Evidence increases that also ion channels of excitable membranes in the heart and the nervous system, including the mammalian brain, can exhibit cooperative properties (Choi, 2014; Molina et al., 2006; Grage et al., 2011; Kim et al., 2014; Gianoli et al., 2017; Navedo et al., 2010; Dixon et al., 2015; Moreno et al., 2016; Clatot et al., 2017; Vivas et al., 2017). Nevertheless, the majority of neuron models relies on the independent gating assumption: channels communicate indirectly via the common membrane potential, but do not directly influence each other. Thus, it is an open question how cooperative channels affect the electrical dynamics - and therefore the computations - of a neuron.

Experimentally, ion channel cooperativity has been studied in various channel types with key roles in the nervous system such as potassium (Molina et al., 2006; Kim et al., 2014), sodium (Iwasa et al., 1986; Undrovinas et al., 1992; Clatot et al., 2017), HCN (Dekker and Yellen, 2006) and calcium channels (Dixon et al., 2015; Moreno et al., 2016). However, most studies have focused on the demonstration of cooperative gating in small ensembles of channels and avoided the complex dynamics of the neuron as a whole. An exception is the study of cooperative calcium channels in rodent hippocampal neurons, where optical control of channel coupling has revealed that the spontaneous firing rate rises when channels cooperate (Moreno et al., 2016). With such control of channel interactions, future experiments have the means to test hypothesis on the function of cooperativity.

So far, a few computational studies predict effects of cooperativity. Along these lines, cooperative sodium channels have been suggested to underlie the rapid initiation and low variability of spiking onset in cortical neurons (Naundorf et al., 2006; Huang et al., 2012; for an alternative mechanism see Yu et al., 2008) and mild cooperative interactions in potassium and calcium channels have been shown to modify the steepness of a channel’s activation curve and modulate neural excitability (Zarubin et al., 2012). Furthermore, cardiac alternans, a pathological condition of unstable heart contractions, has been linked to strong degrees of coupling among cooperative calcium channels (Sato et al., 2018). These studies demonstrate that cooperative channels can significantly alter cellular firing properties with a range of effects from advantageous to pathological depending on the interplay with other currents in the cell.

Here, we show in simulations and mathematical analysis that small clusters of cooperative channels with simple and generic activation dynamics can induce a multistability of the membrane potential and demonstrate that this multistability enables a form of cellular memory. Our central observation is that the mutually enhancing nature of cooperative gating favors joint opening and closing of channels, and, more importantly, results in a hysteresis of their gating behavior. We demonstrate that such cooperative channels - when arranged in clusters and located in membranes with ‘normal’ independent conductances that mediate spiking - can induce graded persistent neural activity: spiking that persists after transient suprathreshold depolarization and represents successive inputs with increasing persistent firing rates. Taken together, we propose that cooperativity of a few ion channels is an elegant way to implement a cell-intrinsic memory directly reflected in a cell’s firing. In principle, ion channel cooperativity could thus efficiently complement network-based mechanisms of persistent activation and thereby contribute to decision making (accumulation of evidence) and working memory. Overall, the emergence of memory in a cluster of coupled, yet in isolation memoryless channels suggests a more general design principle: cooperativity serves to dynamically build functionally rich macrochannels from simpler channels.

In the following, we first dissect the mechanism and compare the gating of independent versus cooperative ion channel clusters. We then add clusters of cooperative channels to a simple neuron model with non-cooperative, spike-generating sodium and potassium channels, proving the ability of ion channel cooperativity to mediate graded persistent activity. Finally, we use the dynamic clamp technique to experimentally endow perirhinal cortex neurons with ‘virtual’ cooperative ion channels and show that stable persistent activity can be robustly induced, rendering ion channel cooperativity an efficient and plausible mechanism for cellular voltage memory.

Although the mechanisms of cooperative interactions among ion channels are still not fully understood on the molecular level, it can be assumed that ion channels need to be in spatial proximity to directly interact and gate in a cooperative manner (Gutkin and Ermentrout, 2006). Matching this assumption, cooperativity is often found in channels that form small clusters (Molina et al., 2006; Choi, 2014; Navedo et al., 2010; Moreno et al., 2016). For our study, we therefore assume that the cooperative channels are distributed in multiple clusters and limit cooperative interactions to channels within the same cluster. In this regard, we deviate from previous theoretical studies, where all-to-all interactions between channels in a membrane have been investigated (Naundorf et al., 2006; Zarubin et al., 2012; Huang et al., 2012). To understand the underlying principles, we begin with a comparison of gating properties in a cluster of cooperative versus non-cooperative ion channels.

Gating of a cluster of cooperative channels

A cluster is assumed to comprise $S$ voltage-gated channels of the same type, whose individual activation curve is a monotonically increasing, yet saturating function of voltage. To ensure the generality of the analysis, we make no further assumptions on other channel properties like their ionic nature and therefore simply refer to them as ‘channels’. Cooperative interactions are chosen to be enhancing, that is the opening of a channel in the cluster increases the opening probabilities of all other $S-1$ channels. This increase in opening probability is implemented phenomenologically by shifting the channels’ activation curves to lower voltages for each new opening of a neighbor. The size of the shift $j$ induced by one new opening is a measure of the coupling between channels in the cluster (Naundorf et al., 2006; Zarubin et al., 2012; Huang et al., 2012). An important determinant for the cluster dynamics is the maximal shift $J=(S-1)j$, which a channel experiences, when all neighbors are in the open state (Figure 1A). Variation of this interaction strength from independent (${\displaystyle J}$ = 0 mV) to strongly cooperative (${\displaystyle J}$ = 70 mV) yields the activation curve of a cluster of $S$ channels. Clearly, an isolated channel and channels in an independent ensemble have the same activation curve. Mild cooperative interactions, however, increase the channel activity and make the activation curve steeper. Beyond a critical coupling, the activation curve ‘bends over’, which characterizes the regime of strong cooperativity (Figure 1B).

Figure 1

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Table 1

Figure	$N$	$S$	$J$	$j$	$V_{0.5}$	k	$\tau$	$V_m$	$\sigma$	gcoop	Ecoop
			mV	mV	mV	mV	ms	mV	mV	pS	mV
Figure 1 (independent)	1	6	0	0	−1	15	0.5	−1	30	-	-
Figure 1 (weak coop.)	1	6	22.5	4.5	−1	15	0.5	−1	30	-	-
Figure 1 (strong coop.)	1	6	70	14	−1	15	0.5	−1	30	-	-
Figure 2 (independent)	1	6	0	0	−1	15	0.5	−1	30	-	-
Figure 2 (cooperative)	1	6	100	20	−1	15	0.5	−1	30	-	-
Figure 3	1	5	100	25	−1	15	0.5	−1	30	-	-
Figure 4	100	8	80	11.4	−30	10	120	−30	20	2.5	100
Figure 5	100	8	80	11.4	−30	10	120	−30	20	2.5	100
Figure 6c	90	8	101.5	14.5	−10	15	100	−10	30	1	100
Figure 3—figure supplement 1	1	5	100	25	−1	15	0.5	−1	30	-	-
Figure 4—figure supplement 1A	100	8	80	11.4	−30	10	120	−30	20	2.5	100
Figure 4—figure supplement 1B	100	8	80	11.4	−30	10	120	−30	20	10	−100
Figure 4—figure supplement 2	100	8	80	11.4	−30	10	120	−30	20	2.5	100
Figure 5—figure supplement 1A,B	100	8	80	11.4	−30	10	10–240	−30	20	2.5	100
Figure 5—figure supplement 1C,D	100	8	80	11.4	−30	10	120	−30	20	0.5–10	100
Figure 6—figure supplement 1 (independent channels, red)	90	8	0	0	−10	15	100	−10	30	1	100
Figure 6—figure supplement 1 (cooperative channels, black)	90	8	101.5	14.5	−10	15	100	−10	30	1	100

Simulating a simple voltage clamp experiment in a membrane patch containing a cluster of six channels demonstrates fundamental differences between the gating of independent and cooperative clusters: The conductance dynamics of a cluster of independent channels is identical to the sum of its individual, independent components. Therefore, the single channel opening probability dictates the most probable cluster state (defined by the channel activation curve and the voltage) and the cluster conductance fluctuates around this state following a binomial distribution (Figure 1C, top). At the half-activation voltage, the cluster average is at half of its total conductance. Fluctuations in the total cluster conductance are relatively fast (due to channel gating time constants of ≈ 1 ms).

The picture changes when ion channel cooperativity is introduced. Assuming strong interactions, all channels in a cluster gate open and close quasi-simultaneously (at a voltage level within the regime where the activation curve ‘bends over’). Intuitively, this coordination within a cluster can be easily understood: the opening of ion channels strongly enhances the probability of their neighbors to open. Similarly, the closing of channels reduces the probability of their neighbors to remain open, so that gating dynamics largely synchronize. The distribution of total cluster conductance now deviates from the unimodal, binomial distribution of clusters with independent channels: it becomes bimodal reflecting the alternation between a fully closed and a fully open cluster state (Figure 1C, bottom). Note that the bistability is accompanied by a novel slow time scale; the cluster switches between the all-open or all-closed states at a drastically reduced rate (6 Hz) compared to its constituting channels when independent (1000 Hz). Effectively, the bistability and the slow switching let the cluster of cooperative channels appear as one slow macrochannel (Blunck et al., 1998).

Hysteresis and bistable gating

The new, slow timescale also changes the response of the cluster to a pulsatile stimulation protocol. During a sufficiently strong depolarizing pulse, all channels in both the cooperative and independent cluster open. Once the pulse is over and voltage returns to baseline, the cluster of independent channels follows and channels swiftly return to the closed state. In the case of the cooperative cluster, however, channels in the cluster remain open, despite the return of the clamped voltage value to its original value before the pulse onset. The strong channel interactions introduce what we would like to term ‘stickiness’: the cluster ‘sticks’ to its new open state (even when voltage levels are back to ‘normal’) and can only be released from this state by a sufficiently strong hyperpolarization (Figure 2A). In other words, cooperative gating induces hysteresis and hence allows channels in a cluster, which are fast switching in isolation, to represent (i.e. ‘remember’) voltage values that occurred multiple hundred milliseconds ago.

Figure 2

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The hysteresis in gating arises from the bistability of the cluster. Before the depolarizing pulse, the channels in the cluster are closed, which is a stable cluster state, because the single channel is not activated at this membrane potential and there is no facilitation between closed channels. During the pulse, channels open, and this open cluster configuration is stable at the baseline voltage, because the strong coupling allows the channels to secure each other in the open state. Only when the voltage is lowered further, it overcomes the mutual facilitation and the channels close again.

The bistability reflects those effective cluster activation curves that ‘bend over’ (Figure 2B). Consequently, cluster size, channel coupling and the shape of the isolated channel’s activation curve determine the voltage range of bistability. Both an increase in channel coupling $j$ and cluster size $S$ can lead to a stronger overall cooperativity strength $J$, which is the product of the two. Bistability arises from a critical overall cooperativity strength $J_{crit}$, which depends on the shape of the single channel activation. With increasing $J$, the bistable range broadens and extends to more hyperpolarized voltages (Figure 2C). Correspondingly, for a cluster with more channels and/or strong coupling, which both result in a larger $J$, the voltage pulse to close the cluster has to be stronger in order to leave the bistable range.

Due to their self-excitatory effect, channels in a cooperative cluster can be compared to neurons in a recurrently connected network: if the external input is low, the neurons are silent, but during a short stimulation they switch to a very active state, which they sustain after the stimulation by recurrent self-excitation. Such networks have been proposed to underlie the short-term storage of stimuli in working memory (Durstewitz et al., 2000). This poses the question, whether in a similar way, a cluster of cooperative channels can act as a memory unit for a single neuron. In order to address this question, we first need to better understand the emerging slow timescale of switches between cluster states in the cooperative regime.

Prolonged lifetimes of the open and closed cluster state

As we have seen, hysteresis allows clusters to act as a voltage memory. After high voltages, they remain open and after low voltages, they remain closed. Eventually, however, a cluster spontaneously switches its state - in a quasi-synchronous fashion all channels open or close (Figure 3A). These switches originate in channel noise, stochastic gating of individual channels. If multiple channels coincide in their spontaneous gating, their neighbors quickly follow. Both opening and closing of channels spread by cooperative coupling; in a spiral of facilitation build-up or, reversely, in a spiral of facilitation loss (Figure 3—figure supplement 1). These gating avalanches, however, cannot be triggered by single or few channels, so that the life times of a cluster can be orders of magnitude longer than those of the single channel.

Figure 3 shows the lifetimes of the open and closed states - exemplary in a cluster with five channels - as a function of voltage. These lifetimes are defined as the time ${\tau }_{O\to C}$ it takes, once all channels are open until all channels are closed again, and vice versa for ${\tau }_{C\to O}$. In the center of the bistable range, both open and closed cluster state are very robust against channel fluctuations with mean residence times in the range of seconds (Figure 3A). However, at the verges of the bistable range, channel noise corrupts the stability. At low voltages, an open cluster switches back to the closed state within ≈ 10 ms, whereas at high voltages, within in the same time, a closed cluster spontaneously opens. Thus, for a persistent representation of recent voltage history, the middle voltage range offers the longest memory lifetimes: here both the probability to lose an open cluster state and to lose a closed cluster state are low - a prerequisite for encoding of previous voltages in a persisting cluster state (Figure 3A). The lifetimes in this voltage range define the memory timescale ${\tau }_{max}$ of the cluster, that is the time during which a cluster of cooperative ion channels can act as a reliable memory device. For completeness, Figure 3B shows the numerically derived center of the bistable range $V_{max}$ and the memory timescale ${\tau }_{max}$ for clusters of different sizes and with different channel-channel coupling $j$. We find that both, larger cluster size and stronger channel coupling, favor bistability. Although quantitative data on cooperative interactions is scarce, we note that for a cluster of eight channels coupled with strength $j$ = 17 mV, the mid-voltage of the bistable range decreases to −60 mV and the average lifetime of the open and closed state exceeds hundreds of seconds, rending this paradigm suitable for persistent activation in the context of short-term memory.

In general, our analysis shows that, as expected, larger and more strongly coupled clusters have a more hyperpolarized operation voltage and longer life times. Even for clusters with the same maximal shift $J$ (that is hyperboles in Figure 3B), we observe a pronounced increase in lifetime with cluster size. In other words, large clusters of weakly coupled channels tend to be more stable than small clusters of strongly coupled ones. This observation is explained by a reduction of effective channel noise when more channels are added to a cluster.

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Clusters as cellular memory units can mediate persistent spiking

The analysis so far showed that clusters of cooperative ion channels can implement a memory of recent voltage levels in their opening state. Next, we demonstrate how this property - when exhibited by small clusters of channels not even directly involved in the generation of spikes - can lead to persistent neuronal activity that does not require further network input. Interestingly, we find that clusters of cooperative channels can even solve the computationally harder problem required for a graded form of persistent activity. In this graded form, the neuron can signal previous input strength and input duration in the frequency of long-term stable firing. Hence, we propose that ion channel cooperativity offers a generic mechanism for cellular memory.

Neural signaling events lead to controllable, persistent cluster switches

To this end, we extend a Hodgkin-Huxley type neuron model including (independent) sodium and potassium channels by a set of 100 small clusters of cooperative channels with the generic activation kinetics described above (for details on the neuron model, see Materials and methods). On the ionic nature of the latter, we assume for the moment that they conduct a depolarizing current like for example calcium channels, which are a potential candidate (Moreno et al., 2016). For the small number of cooperative channels, we simulate their stochastic gating, whereas for the other ionic currents, we use a deterministic simulation.

Stimulating the model cell with transient input pulses elicits spiking at a rate that increases with the stimulus amplitude (Figure 4). Correspondingly, the stimulation amplitude determines the response of the cooperative channels: At low input amplitudes (and consequently low ensuing firing rates), few channels in the clusters open transiently, but close again rapidly (Figure 4A, top trace). When the pulse is over, the cell falls silent and does not generate spikes. In contrast, if the firing rate during the pulse is large enough, in a subset of clusters all channels open. Moreover, in agreement with our above observations, these clusters now remain open when the input pulse is over (Figure 4A, middle and bottom traces). This additional depolarizing conductance suffices to keep the cell in a firing modus and the model neuron exhibits firing activity that persists beyond the presentation of the stimulus pulse.

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Furthermore, the larger the amplitude of the stimulating pulse, the larger the persistent firing rate following the pulse (owing to the persistent opening of more clusters). As every cluster of cooperative channels has a stable open and closed configuration, the population conductance across all clusters allows for a quasi-continuous range of persistent currents. When all cluster are open, the maximal current is reached and the persistent firing rate saturates (≈ 15 Hz). According to the gating properties of cooperative channel clusters, sufficiently hyperpolarizing pulses should be able to close clusters and hence ‘reset’ the memory of the previous pulse by annihilation of the persistent activity. Indeed, Figure 4B shows that a hyperpolarizing pulse cancels persistent activity.

In order to ensure maximal stability when there are no inputs to the neuron, the center of the cluster bistability (−70 mV) coincides with the resting potential (−67 mV). The limited extent of the bistable range from around −90 mV to −50 mV allows the clusters to react to spikes or strong inhibition. For the occurrence of graded levels of persistent activity, we choose channels that do not open instantaneously during a spike. Otherwise, a single spike would open all clusters at once. In contrast, for slower channels with a time constant exceeding the width of a single action potential (>10 ms), only a small ratio of the closed channels opens per spike, so that the conductance gradually increases. The channels considered here are even more inert (≈ 100 ms), comparable to slow adaptation currents that require multiple spikes to fully develop (Benda and Herz, 2003). Similarly, clusters of inert channels require multiple spikes in fast succession to switch to the open state (minimum of ≈ 20 Hz, see Figure 4C). In this way, persistent activity at low frequencies cannot open further clusters and remains stable. Similarly, we find that the slow channels make the clusters robust against fluctuation-driven firing (Schreiber et al., 2009) - action potentials triggered by noise in the membrane potential (see Figure 4—figure supplement 2). Finally, the cell is slightly depolarized, so that it spikes in the absence of a stimulus when several clusters are open (see Figure 4C). Even when there are too few open clusters to induce spiking, they still increase the membrane potential and leave a form of ‘silent’ memory (not shown).

Interestingly, the bistable clusters also support an ‘inverse’ form of neuronal memory - persistent activity activated by hyperpolarization and silenced by strong depolarization. For this other form, the cooperative channel dynamics are the same, but they are required to conduct a hyperpolarizing current like for example potassium channels. Then, open clusters act as an additional standing leak current and prevent the cell from firing. Permanently closing the clusters by a hyperpolarizing pulse relieves the cell from this additional leak and allows persistent firing. However, when the cell is stimulated to spike strongly, the clusters return to the open state, restore the standing leak and again silence the cell (see Figure 4—figure supplement 1). Similarly to depolarization-activated persistent activity, the memory relies on the cooperative dynamics of the channels, but how the neuron makes use of this memory changes with their ionic nature.

Cooperative channels mediate graded persistent activity

Graded persistent activity is an intriguing, cell-intrinsic feature of some neurons that has been reported in vitro under carbachol in the entorhinal cortex, the perirhinal cortex and the amygdala (Egorov et al., 2002; Navaroli et al., 2012; Egorov et al., 2006). These neurons fire at high frequency in response to depolarizing current pulses and continue to fire stably when the stimulus is over. The persistent firing rate is lower than the one during the pulse and increases with each new pulse presented. Hyperpolarizing pulses lead to graded decreases in firing rate and finally end the persistent activity, seemingly resetting the cell. While other mechanisms reproducing this firing mode have been suggested in the literature (Loewenstein and Sompolinsky, 2003; Fransén et al., 2006), we here add a novel candidate and show that cooperative clusters allow to capture essential properties of graded persistent activity.

In analogy to the experimental paradigm used by Egorov et al. (2002), our neuron model was subjected to a series of depolarizing current pulses. Figure 5A shows the persistent firing after the first pulse as well as the increase of the frequency after each pulse. The persistent frequency was stable for at least one minute (data not shown). The role of the clusters as the underlying memory variable becomes apparent in the step-like evolution of their conductance, which increases during the pulses and stays constant in between. In this simulation, we observe four distinct stable persistent frequencies (from about 3 Hz to 10 Hz). At 10 Hz, further depolarizing pulses fail to increase the frequency of persistent firing. This saturation is reached, when all clusters have been switched to the open state. In addition, we confirmed that persistent firing can be gradually turned off by hyperpolarizing pulses (Figure 5B).

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We further investigated which other channel properties - next to strong cooperativity - are essential for the graded nature of persistent activity (Figure 5—figure supplement 1). We concentrated on the time constant and the conductance of the channels. Fast channels, as discussed in the previous section, are less suited than slow channels. The slow channels ensure that clusters gradually open during stimulated fast spiking and halt during slow spiking, so that the persistent firing remains stable. The conductance of the channels has to be at an appropriate intermediate level: sufficient to drive the neuron over threshold, but limited to provide persistent firing frequencies of maximal 10–15 Hz. Otherwise, faster spiking can open further clusters and the persistent activity increases, so that it is no longer stable.

Concluding, a small set of cooperative clusters of ion channels - when combined with ‘normal’, non-cooperative potassium and sodium channels (responsible for the generation of action potentials) - can reproduce the computationally relevant feature of graded persistent activity.

Cooperative channels induce a switch to mnenomic firing in perirhinal cortex neurons

Last but not least, we test the cooperativity mechanism experimentally. Our simulations indicate that clusters of cooperative ion channels can enable the mnemonic firing mode of graded persistent activity in a simple point neuron. In biological neurons, however, other ionic currents, morphological effects or noise could interfere with the action of the cooperative channels and undermine their role as memory units. Thus, we employed the dynamic clamp technique (Sharp et al., 1993; Prinz et al., 2004) to equip perirhinal cortex (PR) neurons in-vitro with the same cooperative clusters used in our models, acting as ‘virtual channels’ in real cells (Figure 6). PR neurons have been shown to exhibit graded persistent activity under activation of muscarinic acetylcholine receptors (Navaroli et al., 2012). Consequently, we applied no carbachol, so that any form of persistent activity observed in the recorded PR neurons stems from the clusters of cooperative channels introduced via dynamic clamp. Accordingly, all recordings were performed in the presence of a synaptic blocker to exclude network contributions. For details of experiments, see Materials and methods.

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The dynamic clamp technique allowed us to control the cluster size, channel interactions and channel kinetics of the artificial conductances during the experiment. Like in the model neuron simulations, we adapted the cluster parameters so that the bistability range fitted to the properties of the recorded cell.The resting potential determined the center of the bistable range, whereas the spike shape set boundaries for the bistable range. Voltage values during afterhyperpolarization lied above the lower boundary (to prevent reset by a spike’s afterhyperpolarization), but the upstroke of the action potential exceeded the upper boundary of the bistable range.

Stable recordings with cooperative channel dynamics added via dynamic clamp were obtained for three PR neurons. Under cooperative conditions, all cells exhibited persistent activity in response to current stimulation with pulses (Figure 6C). Persisting activity following pulses was graded, that is the persistent firing rate increased with each presented pulse. As in the mathematical model, hyperpolarizing pulses lowered the rate of persistent activity and when applied multiple times, silenced it (Figure 6D). In the control conditions (identical, yet non-cooperative channel clusters), no persistent firing could be evoked (Figure 6—figure supplement 1).

Periods of stimulation were followed by a slow afterhyperpolarization (sAHP) in both control and cooperative conditions, which for paradigms with cooperative gating generated a 3–4 s lasting refractory period between the stimulation and the self-sustained firing. Interestingly (and in agreement with mathematical modelling), the persistent activity ‘unfolded’ despite these intrinsically generated dynamics. We note that such sAHP were not expected in the original recordings of carbachol-induced persistent activity (Egorov et al., 2002), as carbachol blocks sAHPs.

Taken together, the experimental data support the hypothesis that cooperative ion channels can mediate cellular persistent activity. The persistent dynamics are very robust and could be easily evoked without further knowledge of the intrinsic properties of the recorded cells, such as the multitude of other ionic currents, complex cell morphology, and the presence of channel noise.

Despite a large body of evidence for cooperative interactions between channels pivotal to the nervous system (Iwasa et al., 1986; Dekker and Yellen, 2006; Kim et al., 2014; Choi, 2014; Dixon et al., 2015; Moreno et al., 2016; Clatot et al., 2017), it is unknown which function coupled channels have for neural dynamics and computation. Based on a mathematical analysis, we show in this study that a cluster of cooperative channels can gate with hysteresis and find, in both simulations and experiments, that multiple such clusters embedded in the membrane of a neuron mediate mnemonic firing like persistent activity. Therefore, we suggest that ion channel cooperativity might serve as a cell-intrinsic memory mechanism at the voltage level.

From synaptic learning, adaption of the immune system to epigenetics, many biological memory systems are founded on strong positive autofeedback (Lisman, 1985; Burrill and Silver, 2010). Feedback loops are ubiquitous in biology (Thomas and D’Ari, 1990); they amplify signals as in calcium-induced calcium release and enhance sensitivity in controlling protein function by multisite phosphorylation. In particular, strong feedback enables bistability and memory as in networks of recurrently connected neurons (Amit, 1990; Durstewitz et al., 2000). Here, we exploit the same principles for voltage-gated ion channels, where cooperativity can act as a strong auto-feedback and enables clusters of channels to act as bistable macrochannels with a memory of previous voltage levels.

The model for the cooperative ion channels and the neuron are described in the manuscript. Software for simulation of the cooperative channels is provided in git repositories linked in the manuscript (https://itbgit.biologie.hu-berlin.de/cooperativity/brutelib; copy archived athttps://github.com/elifesciences-publications/49974-brutelib; and https://itbgit.biologie.hu-berlin.de/cooperativity/dynamic_clamp_model; copy archived at https://github.com/elifesciences-publications/49974-dynamic_clamp_model). The recordings and analysis obtained in the dynamic clamp experiments are explained in the manuscript and are publicly available at https://gin.g-node.org/doi/cooperative_channels_in_biological_neurons_via_dynamic_clamp.

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Author details

1. Paul Pfeiffer

   1. Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Berlin, Germany
   2. Bernstein Center for Computational Neuroscience, Humboldt-Universität zu Berlin, Berlin, Germany

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Funding acquisition, Validation, Investigation, Visualization, Methodology

   For correspondence

   pfeiffpa@hu-berlin.de

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-5324-5886

2. Alexei V Egorov

   Institute of Physiology and Pathophysiology, Heidelberg University, Heidelberg, Germany

   Contribution

   Supervision, Investigation, Methodology

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-4899-8407

3. Franziska Lorenz

   Institute of Physiology and Pathophysiology, Heidelberg University, Heidelberg, Germany

   Contribution

   Data curation, Software, Investigation, Methodology

   Competing interests

   No competing interests declared

4. Jan-Hendrik Schleimer

   1. Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Berlin, Germany
   2. Bernstein Center for Computational Neuroscience, Humboldt-Universität zu Berlin, Berlin, Germany

   Contribution

   Conceptualization, Software, Formal analysis, Supervision, Validation, Methodology

   Competing interests

   No competing interests declared

5. Andreas Draguhn

   Institute of Physiology and Pathophysiology, Heidelberg University, Heidelberg, Germany

   Contribution

   Resources, Supervision, Funding acquisition, Methodology, Project administration

   Competing interests

   No competing interests declared

6. Susanne Schreiber

   1. Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Berlin, Germany
   2. Bernstein Center for Computational Neuroscience, Humboldt-Universität zu Berlin, Berlin, Germany

   Contribution

   Conceptualization, Resources, Formal analysis, Supervision, Funding acquisition, Investigation, Visualization, Methodology, Project administration

   For correspondence

   s.schreiber@hu-berlin.de

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-3913-5650

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