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Local changes in potassium ions regulate input integration in active dendrites [1]

['Malthe S. Nordentoft', 'Niels Bohr Institute', 'University Of Copenhagen', 'Copenhagen', 'Naoya Takahashi', 'University Of Bordeaux', 'Cnrs', 'Interdisciplinary Institute For Neuroscience', 'Iins', 'Umr']

Date: 2024-12

During neuronal activity, the extracellular concentration of potassium ions ([K + ] o ) increases substantially above resting levels, yet it remains unclear what role these [K + ] o changes play in the dendritic integration of synaptic inputs. We here used mathematical formulations and biophysical modeling to explore the role of synaptic activity-dependent K + changes in dendritic segments of a visual cortex pyramidal neuron, receiving inputs tuned to stimulus orientation. We found that the spatial arrangement of inputs dictates the magnitude of [K + ] o changes in the dendrites: Dendritic segments receiving similarly tuned inputs can attain substantially higher [K + ] o increases than segments receiving diversely tuned inputs. These [K + ] o elevations in turn increase dendritic excitability, leading to more robust and prolonged dendritic spikes. Ultimately, these local effects amplify the gain of neuronal input–output transformations, causing higher orientation-tuned somatic firing rates without compromising orientation selectivity. Our results suggest that local, activity-dependent [K + ] o changes in dendrites may act as a “volume knob” that determines the impact of synaptic inputs on feature-tuned neuronal firing.

Funding: R.N.R. acknowledges support from the Lundbeck Foundation (R370-2021-764). M.S.H acknowledges support from the Lundbeck Foundation and (R347-2020-2250) and the Novo Nordisk Foundation (NNF23OC0085907). M.H.J. acknowledges support from the Independent Research Fund Denmark (9040-00116B) and the Novo Nordisk Foundation (NNF20OC0064978 and NNF24OC0089788). M.S N. acknowledges support from Novo Nordisk Foundation (NNF20OC0064978). A.P. acknowledges support from the Theodore Papazoglou FORTH Synergy Grants. N.T. acknowledges support from French National Centre for Scientific Research (CNRS), the framework of the University of Bordeaux’s IdEx “Investments for the Future” programs (2020 IdEx Junior Chair; GPR BRAIN_2030), Conseil régional Nouvelle-Aquitaine (Bordeaux Neurocampus Junior Chair), the ATIP-Avenir program, Fondation Schlumberger pour l’Education et la Recherche (FSER202401018842), Brain Science Foundation, and Research Foundation for Opto-Science and Technology. None of the sponsors or funders played any role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Copyright: © 2024 Nordentoft et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

To test the hypothesis, we used proof-of-concept mathematical formulations and biophysical modeling to investigate the effect of the reported physiological [K + ] o changes. We focus on the organization of excitatory synaptic inputs tuned to the stimulus orientation on dendritic branches of a pyramidal neuron in the visual cortex [ 42 ], as we have previously experimentally shown that visual cortical responses are dynamically regulated by [K + ] o and brain state [ 14 ]. Using statistical analysis, as well as abstract and morphologically detailed biophysical models of dendrites and neurons, we show that the arrangement of tuned inputs determines the magnitude of activity-dependent [K + ] o changes in dendrites. Specifically, dendritic segments with similarly tuned synaptic inputs can attain substantially higher [K + ] o elevations than segments with diversely tuned inputs. These [K + ] o elevations in turn depolarize the E K + which increases the reliability of dendritic spikes and prolongs their duration, but without compromising their stimulus selectivity. Ultimately, these local effects amplify the gain of neuronal input–output transformations, leading to higher firing rates at the soma without affecting feature selectivity. Overall, our results suggest a prominent role for local, activity-dependent, dendritic “[K + ] o hotspots” [ 21 , 27 ] in shaping dendritic integration of synaptic inputs. Importantly, using the visual cortex as a framework of study and grounded by experimental data, we provide a proof-of-concept that grouping of co-tuned inputs creates dendritic [K + ] o hotspots that regulate dendritic integration, a property can be generalized to other sensory features, brain regions, or animal species, with the requirement that synaptic inputs are clustered on a spatial and temporal scale [ 32 , 42 , 45 – 50 ].

Pyramidal neurons possess elaborate dendritic trees that contain a variety of voltage-dependent ion channels such as Na + , K + , and Ca 2+ channels. These channels allow for complex, active responses to synaptic inputs, including the generation of local action potentials, called dendritic spikes [ 28 – 31 ]. The spatial arrangement of synaptic inputs is an important factor for determining dendritic spike initiation [ 32 – 38 ]. These dendritic spikes support the nonlinear summation of synaptic inputs, which in turn alter the neurons’ input–output function and can enhance sensory feature selectivity in vivo [ 39 – 42 ]. Interestingly, artificial, pharmacological manipulation of dendritic K + currents affects dendritic excitability and dendritic spikes, indicating that K + currents can act as a regulator of dendritic integration [ 43 , 44 ]. Based on the above findings, we here hypothesize that physiological, synaptic activity-mediated changes in [K + ] o can locally regulate the active dendritic properties and thus shape the nonlinear integration of inputs and sensory processing.

Throughout the nervous system, neuronal activity and ionic changes in the extracellular environment are bidirectionally linked. Yet, extracellular ion changes are not traditionally considered an integral part of neuronal signaling and information processing. Amidst the activity-dependent fluctuations of extracellular ionic concentrations, K + ions emerge as particularly intriguing due to their pivotal role in shaping neuronal excitability and membrane potential (V m ). At rest, the extracellular concentration of K + ([K + ] o ) in the brain is normally between 2.7 and 3.5 mM [ 1 – 3 ]. It has been experimentally shown that during sensory stimulation, motor network activity, sleep oscillations, or behavioral state transitions, [K + ] o increases by 0.25–2 mM [ 4 – 14 ], while it can rise up to 7–12 mM during hypersynchronous neuronal activity [ 15 – 17 ]. The [K + ] o increase weakens the outward K + driving force, resulting in a less negative K + reversal potential (E K + ), which powerfully affects the V m , excitability, and firing patterns of neurons [ 11 , 14 , 18 – 25 ]. A major source contributing to the [K + ] o changes at the synaptic level is K + efflux from excitatory glutamatergic receptors, and in particular from NMDA receptors, when compared to other calcium- or voltage-dependent potassium channels [ 16 , 21 , 26 ]. Importantly, previous work has shown that such postsynaptic NMDA receptor-mediated K + efflux is highly localized, and can signal to presynaptic axons [ 21 , 27 ], pointing to local [K + ] o changes acting as a modulator of presynaptic transmission. Despite the experimental evidence that synaptic changes can be highly localized, we still lack the experimental tools to systematically disentangle the effect of activity-dependent [K + ] o changes at such a fine-scale level. In addition, whether the activity-mediated [K + ] o changes can modulate postsynaptic integration of synaptic inputs remains elusive.

Finally, we asked what functional benefits there might be to this K + -mediated gain modulation. A major task of dendrites is to transmit incoming inputs to the soma for output generation. This is an inherent challenge, especially for distal inputs to large neurons, as the voltage signal tends to attenuate as a function of distance traveled and branching points [ 80 , 81 ]. We, therefore, speculated that one function of K + -mediated gain modulation might be to promote neurons transmitting signals over larger dendritic distances. To test this, we manipulated the synaptic input distance to the soma by scaling the entire neuron while measuring the integrated V m signal at the base of the neuron ( Fig 4f and 4g ). Interestingly, due to the local gain of the synaptic input, the E K + -shifted regime is attenuated less with distance. This, in turn, makes it possible for signals to travel substantially longer, before reaching a similar integrated V m signal. Based on the power-law fits using the χ 2 fitting method for each ΔE K + condition, we observe that as the neuron grows, the signal in the absence of E K + shift is attenuated by a factor when compared to the large E K + shifts ( Fig 4g ). Together, these results show that local activity-dependent [K + ] o increases and E K + shifts in dendrites enhance the effectiveness of distal synaptic inputs to cause feature-tuned firing of neurons, without comprising feature selectivity.

These simulations revealed that somatic firing rates were significantly amplified when E K + values were shifted locally ( Fig 4c and 4d ; P = 0.007 and P = 0.003 for small and large E K + shifts, respectively, one-tailed Student’s t test, N = 15 simulations). We determined that this form of gain modulation was consistent with a multiplicative transformation ( S7 Fig ; gain coefficient = 1.64 and 2.73 for small and large E K + shifts, respectively), similarly to previously shown in the visual cortex [ 79 ]. This amplification was similar to the one expected by increasing the number of synapses ( S8 Fig ) with the main difference being that E K + modulation closely follows the dendritic activity levels, operating in time scales of seconds. Importantly, and in congruence with what we observed in the dendrites ( Fig 2h ), the somatic firing selectivity was not changed by E K + shifts despite increases in firing rate also for orientations away from the target orientation ( Fig 4c and 4d ; orientation selectivity index: 0.99, 0.98, and 0.98 for no, small, and large E K + shifts, respectively; N = 15 simulations). To elucidate how dendritic E K + shifts cause dendritic gain modulation, we measured the area under the curve of the integrated V m signal arriving at the base of the dendritic trunk during a stimulation event. To exclude back-propagating action potentials (bAPs), for this analysis, we turned off the voltage-gated sodium channels in the soma and trunk. This showed that the strength of the integrated dendritic signal, received by the soma, rose substantially with increasing E K + shifts ( Fig 4e ); for example, the signal strength at the target orientation increased by around 12% when imposing large E K + shifts compared to no E K + shifts.

(a) Diagram of the abstract neuron model. The neuron received orientation-tuned synaptic input at the distal dendrites and local E K + shifts. (b) Example soma V m traces obtained as a function of dendritic E K + shift magnitude and stimulus orientation relative to target orientation. (c) Soma firing rate tuning curve for different dendritic E K + shift magnitudes. Error bars are mean ± SEM. (N = 15 simulations). ***P = 10 −16 and 10 −30 for small and large E K + shifts, respectively, one-tailed Student’s t test. (d) Soma peak firing rate at target orientation and orientation selectivity index as a function of dendritic E K + shift magnitude. Error bars are mean ± SEM. (N = 15 simulations). (e) Area under the curve for the dendritic trunk V m measurements as a function of stimulus orientation relative to target orientation and dendritic E K + shift magnitude. Error bars are mean ± SEM. (N = 15 simulations). Trunk length was kept constant at 500 μm. Note that bAPs were here abolished by silencing voltage-gated sodium channels in the trunk and soma. (f) Example dendritic trunk V m traces obtained at the target orientation as a function of synaptic input distance to soma and dendritic E K + shift magnitude. (g) Area under the curve for the dendritic trunk V m measurements at the target orientation as a function of synaptic input distance to soma and E K + shift magnitude. Error bars are mean ± SEM. (N = 15 simulations) and solid lines represent fit to the function f(x) = x a +c. ***P = 0.61, 0.41, and 0.38 for different E K + shifts. See also S7 Fig and S2 Table .

Our data suggests that the local [K + ] o increases in dendritic segments receiving similarly tuned inputs can boost the reliability and duration of dendritic spikes without compromising feature selectivity ( Fig 2g and 2h ). Thus, we next sought to investigate the functional consequences this result might have on the somatic neuronal output. For this, we constructed an abstract neuron model using a fractal tree, mimicking the generic compartmentalization and fractal dimensionality of the apical dendritic tree of pyramidal neurons [ 75 , 76 ] and we stimulated distal dendrites with similarly tuned input [ 38 , 77 , 78 ] ( Fig 4a ). We simulated 3 conditions of E K + shifts in the stimulated dendrites: no change, small shift, or large shift, based on the [K + ] o changes determined in Fig 1 ( Fig 4a and 4b ; see Methods ). These 3 levels correspond to ΔE K + = 0 mV, ΔE K + = 6 mV, and ΔE K + = 18 mV, respectively, to represent the case of no contribution as well as the minimum and maximum level used in ( Fig 1 ), based on in vivo recordings. In S6 Fig , we also included a similar test using a reconstructed neuron morphology based on a previously published model.

Shifting the E K + alters the I-V curve attractor landscape in 3 fundamental ways ( Fig 3b–3e and S1 Video ). First, for the down-stable state, it moves the fixed point to a more depolarized V m , as well as reducing the net outward current across all V m levels ( Fig 3b ). Second, for the bistable state, it lowers the voltage barrier needed to transition the system to the up-stable state and deepens the attractor basin ( Fig 3c and 3d ). Finally, by reducing the net outward currents at depolarized V m levels, it prolongs the duration for which the system is in up- and bistable states ( Fig 3c–3e ). These dynamical system properties are all explained by the weakened K + driving force through K + channels and fully predict the ΔE K + -mediated effects we observed on dendritic spiking ( Fig 2 ). Collectively, this analysis demonstrates that the effects of local K + changes on synaptic integration can be predicted by the altered dynamical system properties of dendritic spikes.

(a) I-V curves during the generation of dendritic NMDA spike. Green: down-stable state with only intrinsic ion channels active. Blue: bistable state with intrinsic ion channels and NMDA receptor active. Red: Bi- and up-stable state with intrinsic ion channels and NMDA and AMPA receptors active. (b) Down-stable state before dendritic spike generation without and with E K + shift. Solid points indicate stable fixed points. (c, d) Bistable states during and at the end of dendritic spike without and with E K + shift. (c) Shows the hypothetical system with peak NMDA receptor conductance, without AMPA receptor activation, right before spike initiation, and (d) shows the system when outward and inward currents match for the system without E K + shift around the end of the spike. Solid and open points indicate stable and unstable fixed points, respectively. (e) Heat maps showing the temporal evolution of the I-V landscape without and with E K + shift. The corresponding I-V curves shown in (b–d) are indicated with black dotted lines, and full and dotted blue lines indicate stable and unstable fixed points, respectively. (a–c) Arrows indicate system flow direction. Note that positive inward current depolarizes the membrane, resulting in I–V plot similar to previous work (Major and colleagues [ 29 ]). See also S1 Video .

To understand the general biophysical principles governing the regulation of dendritic spikes by local K + changes, we turned to dynamical systems theory [ 74 ]. Dendritic spiking can be described by the instantaneous phase plot of the V m and . Given that the membrane capacitance is constant, we transformed this to the I-V curve using [ 29 ]. With only the intrinsic ion channels active, V m is attracted to a single down-stable hyperpolarized fixed point corresponding to the resting potential ( Fig 3a ). Inclusion of NMDA receptors changes the system to bistable, and an attractor basin together with a depolarized fixed point, corresponding to dendritic spiking, is created. The voltage barrier preventing the system from transitioning to an up-stable state, where V m is only attracted to the depolarized fixed point, is overcome by the activation of AMPA receptors. As the activity of AMPA and NMDA receptors wears off, in conjunction with the high activity of K + channels, the system is again pushed back to the down-stable state.

Using Eq ( 3 ), we estimated the probability of generating a dendritic spike for a given stimulus orientation ( Fig 2g ). The probability of eliciting dendritic spikes rose drastically as a function of increasing E K + shifts ( Fig 2g and 2f ); for example, the probability of spiking to the target orientation increased from 48% to 97% when shifting the E K + by 12 mV. This gain in the dendritic output was similar to the one expected from increasing the number of synapses ( S5 Fig ), with the main difference being that E K + modulation is transient, selectively boosting the activity of repetitively active dendrites. Interestingly, the dendritic spiking orientation selectivity was largely constant even though the probability of spiking increased also for orientations away from the target orientation ( Fig 2g and 2h ; Orientation selectivity index: 0.98, 0.98, and 0.96 for ΔE K + = 0, 6, and 18 mV, respectively). Altogether, these results show that local K + changes in dendritic segments with similarly tuned synaptic inputs prolong dendritic spikes and boost the probability of generating dendritic spikes without affecting their feature selectivity.

In our model, 3 main parameters control dendritic spike generation: the number of dendritic synapses (N), the average synaptic activity (w), and the ΔE K + (see Methods for details). By simulating different parameter configurations with dendritic spike occurrence as binary output measure, we fitted a function that describes the minimum ΔE K + needed to trigger dendritic spiking given N and w ( S3 and S4 Figs ): (3)

(a) Diagram of the biophysical point-dendrite model. To test the effect of local [K + ] o elevations, we imposed E K + shifts within the interval of 0–18 mV. (b) Example dendrite V m traces for the similar input tuning regime as a function of stimulus orientation relative to target orientation. Individual trials are in gray and average is in black. Shaded regions indicate synaptic stimulation timing. (c) Example V m traces highlighting dendritic spike duration (time above –30 mV) as a function of ΔE K + at the target orientation. The E K + shift is induced following the first simulation event and is visible in the resting membrane potential of the dendritic segment. (d) Dendritic spike duration as a function of ΔE K + at the target orientation. Error bars are mean ± SEM. (N = 10 simulations). (e) Example traces highlighting V m response as a function of ΔE K + and mean synaptic activity. (f) Heat map showing peak V m depolarization as a function of ΔE K + and mean synaptic activity. Dotted lines indicate the transition to dendritic spiking. Number of synapses used: N = 10. (g) Dendritic spike probability as a function of ΔE K + and stimulus orientation relative to target orientation. Note that ΔE K + values denote shifts at the target orientation and shifts for the rest of the orientations are calculated using the synaptic activity factor. (h) Dendritic spike probability at the target orientation (left axis) and orientation selectivity index (right axis) as a function of ΔE K + . See also S3 and S4 Figs and S1 Table .

We then investigated the implications of the identified in Fig 1 E K + shifts might have on dendritic synaptic integration. We focused on dendritic segments with similarly tuned inputs as (1) we documented stimulus orientation-dependent and large (above 15 mV) E K + shifts in these segments ( Fig 1g and 1h ); and (2) previous work has shown that spatial clustering of co-active inputs is an important factor for dendritic spike initiation [ 32 – 38 ]. For this analysis, we implemented a biophysically detailed point model to simulate the V m of a dendritic segment during synaptic input stimulation, referred to as a “point-dendrite” model due to its similarities with a conventional point-neuron model [ 70 ] ( Fig 2a ; see Methods ). The point-dendrite model included an array of active ion channels found in the dendrites of V1, as well as AMPA and NMDA receptors [ 71 – 73 ] ( S1 Table ). The number of synapses and their orientation tuning were sampled as in Fig 1 to simulate a dendritic segment receiving similarly tuned inputs. Synaptic inputs were activated by delivering a stimulation train consisting of 3 events in which AMPA and NMDA receptors were activated with a peak current depending on the synapse’s orientation tuning and with a Poisson-distributed delay (mean = 80 ms). This protocol induced V m dynamics that reflected the stimulus orientation in the absence of E K + shifts ( Fig 2b ). Importantly, for orientations close to the target orientation, we also observed regenerative activation of NMDA conductances, leading to NMDA-dependent dendritic spikes (see Fig 3a ), while for the orthogonal orientation the V m did not change ( Fig 2b ) [ 42 ]. To obtain a mechanistic understanding of how E K + shifts might shape orientation tuning, we first tested their effect on dendritic spike properties (see Methods ) for a stimulus presented at the target orientation, using different ΔE K + values in the predicted range of Fig 1h , ([0–18] mV). E K + shifts caused a broadening of dendritic spikes ( Figs 2c , 2d and S1 ); for example, for a 12 mV E K + shift the duration of dendritic spikes increased by approximately 80% compared to no E K + shift. This effect persisted in the presence of nonspecific inhibitory (GABA A ) input ( S2 Fig ). Furthermore, the amount of excitatory synaptic drive needed to transition the dendritic V m from subthreshold input summation to suprathreshold dendritic spiking decreased substantially with increased E K + shifts ( Fig 2e and 2f ).

Focusing on activity for stimuli presented at Δtarget orientation = 0°, we next identified the Δ[K + ] i and R 2 parameters that constrained the Δ[K + ] o for the diversely tuned inputs to a range of 0.25–1 mM ( Fig 1h , left ), to reflect the changes measured by in vivo [K + ] o measurements using microelectrodes [ 5 – 7 , 9 , 10 , 12 – 14 , 18 , 65 ], a technique which averages the measured concentration in space and time and likely underestimates true local [K + ] o changes [ 66 – 69 ]. This set of parameters when applied to the similarly tuned input regime results in Δ[K + ] o changes in the range 1.25–5 mM ( Fig 1h , right ). By converting the Δ[K + ] into shifts in E K + via the Nernst equation: (2) and using this parameter set, we evaluated the range of the ΔE K + shifts. For diverse inputs, the ΔE K + is within the range of 1.6–5.9 mV above the resting E K + , and for similarly tuned input, the respective range is 6–18 mV ( Fig 1h ). Based on this analytical approach, in the following sections, we use these ranges in ΔE K + as a proxy for changes in [K + ] o , when we explore their impact on dendritic integration in biophysical models with different levels of abstraction. Together, these data suggest that dendritic segments receiving synaptic inputs with similar tuning preferences can attain substantially higher [K + ] o and E K + changes compared with segments receiving diversely tuned inputs, yet in a stimulus-specific manner.

Here, Δ[K + ] o and Δ[K + ] i are the changes in [K + ] o and [K + ] i , respectively. The dendritic segment is considered as a cylinder with radius R 1 , encapsulated by the extracellular space, also described as a cylinder with radius R 2 (see Methods , Eq 5 ). By keeping the dendritic diameter constant (R 1 = 1 μm) [ 60 , 61 ], we limit our parameter space to 2 parameters: Δ[K + ] i and R 2 . Although we do not have data on the local dendritic [K + ] i changes, nor do we know the exact extracellular space size in the vicinity of dendritic segments, we can use Eq 1 to estimate [K + ] o changes by varying Δ[K + ] i and R 2 within realistic ranges [ 62 – 64 ]. Note that as R 2 →R 1 , Δ[K + ] o increases as approaching infinity, and becomes undefined for R 2 = R 1 . As R 2 →∞, Δ[K + ] o goes towards zero asymptotically. By multiplying Δ[K + ] i for the diversely tuned input regime by the synaptic activity factor ( Fig 1f ), we also obtain relative estimates of [K + ] o changes in segments receiving similarly tuned inputs. Δ[K + ] o changes are positively correlated with the Δ[K + ] i ( Fig 1g , right ), inversely correlated with the size of the extracellular space ( Fig 1g , left ) and, for the similar input regime, higher for orientations close to the target orientation but lower for orientations far from the target ( Fig 1g ).

We then asked how these synaptic activity patterns are manifested in [K + ] o and E K + changes. Due to experimental limitations, there is a lack of knowledge about singular ionic flux over the multiple ionic channels that support the in vitro and in vivo documented potassium concentration changes. Thus, we adopted a mean-field approach that allowed us to relate the experimental recordings of extracellular potassium to the model while keeping to the number of assumptions at a minimum. Specifically, in our approach we assumed the following conditions to be true: (1) Local increases in [K + ] o are proportional to synaptic activity levels; (2) local increases in [K + ] o are caused by K + efflux from the intracellular space; and (3) [K + ] o change is stable in space and time along the short dendritic segment. The latter is motivated by the fact that (a) K + ions in the extracellular space, of the spatial and temporal scale considered here (approximately 10 μm and approximately 300 ms after synaptic activity onset, respectively), can be described as well-mixed due to the high K + free diffusion rate ( ) [ 56 , 57 ]; and (b) extracellular potassium remains elevated for timescales of several hundreds of ms, as documented in in vitro experiments [ 58 , 59 ] (see Methods for detailed analysis). Following these assumptions, we approximated the [K + ] o changes of dendritic segments by multiplying the intracellular K + concentration ([K + ] i ) change by the volume fraction between the intra- and extracellular space: (1)

(a) Diagram of the proposed hypothesis that dendritic segments with similarly tuned inputs can attain higher [K + ] o increases than segments with diversely tuned inputs. (b) Cumulative fraction of synapse orientation preferences relative to target orientation for dendritic segments with diverse or similar orientation tuning. Reproducing results of [ 42 ]. (c) Example dendritic segments with diverse and similar input tuning regimes (left). Synaptic tuning curves for different orientation-tuned stimuli, as per [ 55 ], normalized by the maximum activity (right). Note that the x-axis corresponds to stimulus orientation relative to target orientation. The resulting activity shows the tuning curves for the synapses indicated on the corresponding dendritic segment on the left. Based on the distributions of panel b and the synaptic tuning curves, we acquired a statistical approximation of the expected E K + shifts without explicitly modeling membrane dynamics. (d) Example synapse-averaged tuning curves for dendritic segments from the diverse (top) and similar (bottom) tuning regimes. Tuning curves of individual synapses are in gray and average is in color. (e) Probability distributions of average synaptic activity for the diverse and similar tuning regimes for the target (left) and orthogonal to the target (right) orientations. Dotted lines indicate expectation value. (f) Synaptic activity factor, defined as the ratio between expected synaptic activity for similarly— E (S) and diversely— E (D) tuned segments as a function of stimulus orientation relative to target orientation. (g) Δ[K + ] o as a function of stimulus orientation relative to target orientation for the diverse and similar tuning regimes. Left: different extracellular space radii (R 2 ) with constant [K + ] i reductions (Δ[K + ] i = 1 mM). Right: different Δ[K + ] i with constant R 2 ( ). (h) Heat maps showing the range of ΔE K + for the diverse (left) and similar (right) tuning regimes as a function of Δ[K + ] i and R 2 at the target orientation. Dotted lines indicate the corresponding Δ[K + ] o range.

To identify the expected E K + shifts under different conditions, we first undertook an analytical approach that did not include simulation of the membrane potential dynamics. We hypothesized that dendritic segments with similarly tuned inputs can attain higher [K + ] o changes and E K + shifts than segments with diversely tuned inputs and that this could, in turn, affect dendritic excitability ( Fig 1a ). To investigate this, we sampled the synaptic activity of a hypothetical 10 μm segment, populated with 7 to 13 synapses, yielding a synaptic density comparable to the neocortex [ 51 – 53 ]. We chose this length scale because functional clusters, comprised of inputs with similar tuning preferences, typically are formed within 5 to 10 μm [ 32 , 45 – 47 ]. To model feature-tuned synaptic inputs, we used visual orientation tuning as our framework. Previous work identified dendritic segments receiving inputs with similar or diverse orientation tuning [ 42 , 54 ]. To capture this, we sampled synaptic orientation preferences from a circular normal distribution for the similarly tuned regime and from a uniform distribution for the diversely tuned regime [ 42 ], and assigned tuning curves to individual synapses ( Fig 1a and 1c ). From the sampled segments, we obtained synapse-averaged tuning curves and activity distributions ( Fig 1d and 1e ). Using these, we derived an activity factor for each orientation, signifying the ratio of expected synaptic activity in segments with similar versus diverse tuning preferences ( Fig 1f ; see Methods ). This showed that for stimulus orientations close to the target orientation, here arbitrarily chosen as the somatic preferred orientation set at 0°, (Δtarget orientation: 0–20°), synaptic activity levels are 2 to 5 times higher in segments with similarly tuned inputs than in segments with diversely tuned inputs. Contrary, for orientations far from the target orientation (Δtarget orientation: 45–90°), activity ceases almost entirely in segments with similar tuning while it remains constant for segments with diverse tuning, yielding activity factors below 1.

In conclusion, our study is a first step towards unraveling the role of extracellular K + changes in dendritic integration. Future theoretical and experimental studies are needed to obtain a comprehensive understanding of how local activity-dependent ionic shifts contribute to information processing and computations in dendrites.

Our approach also comes with its limitations: First, while we focused only on the postsynaptic impact of local [K + ] o changes, it is reasonable to predict that activity-dependent [K + ] o hotspots could also affect the presynaptic terminals [ 18 ]. Increases in [K + ] o depolarize axons, which can broaden action potentials and increase presynaptic calcium entry [ 95 , 96 ], leading to enhanced glutamate release and stronger synaptic transmission. Such amplification of local synaptic integration, through pre- and post-synaptic mechanisms, could play important role in neuronal circuit development and long-term potentiation by supporting spike timing-dependent plasticity [ 97 , 98 ]. Second, [K + ] o changes may modulate the gating properties of specific types of K + channels, such as the inward-rectifier K + (Kir) [ 99 ] or the slow delayed rectifier (KCNQ1) [ 100 , 101 ], potentially providing another venue for [K + ] o self-regulating mechanism of dendritic excitability. Third, synapses surrounding the dendritic segments may also participate or be affected by the local [K + ] o changes: activity of the surrounding synapses, belonging to dendritic branches of the same or different neurons may also contribute to the local [K + ] o . This co-activation of nearby dendrites could either increase the noise levels or locally promote the synergy between synapses, dendrites, and neurons. Fourth, in this work, we did not assess in detail GABAergic inhibition, known to play important roles in regulating dendritic activity [ 102 , 103 ]. Future work could address if local [K + ] o increases near dendritic segments with similarly tuned excitatory inputs could cause disinhibition by reducing the inward driving force for chloride ions thus increasing neuronal feature selectivity or creating a temporal window for dendritic plasticity. Fifth, the width of the extracellular space can be highly heterogeneous, ranging from 0.04 to 0.5 μm [ 63 , 104 ], yet smaller than the pyramidal neuron dendritic diameter [ 105 , 106 ]. Importantly, the extracellular space itself is a dynamic compartment with timescales ranging from seconds to hours; for example, the extracellular space decreases in response to epileptiform activity [ 107 ] and these structural changes can be highly local [ 63 ], making the extracellular space size another mean for local modulation of dendritic excitability. Finally, our mean field approximation quantifies average [K + ] o changes after synaptic activation and thus lacks temporal resolution. We chose this approach because several important parameters have yet to be experimentally determined, including the voltage-dependent fraction of K + currents of total NMDA currents and the precise geometry of the extracellular space surrounding functionally mapped dendritic segments, which prevents precise simulations of time-dependent [K + ] o dynamics. We assumed here that the system goes asymptotically towards the well-mixed state following synaptic activation, and we estimate that this is reached approximately 150–170 ms after input onset, but we cannot infer the nature of the spatiotemporal [K + ] o dynamics before this time point. In the future, experimental knowledge about these parameters would enable us to make a full-scale electrodiffussive model of the extracellular space, improving the temporal resolution of extracellular ion dynamics.

Importantly, our prediction on the existence of the local activity-dependent [K + ] o hotspots can be potentially experimentally tested. The gold standard for measuring [K + ] o dynamics in the brain is with K + -selective microelectrodes [ 5 , 7 – 10 , 12 – 14 , 65 ]. However, this technique creates a dead space surrounding the electrode, and can only measure from a single point in space, hence providing poor spatial resolution. Instead, to test our hypothesis, an optical approach like two-photon microscopy with fine-scale synaptic resolution seems ideal. Genetically encoded green fluorescent protein-based [K + ] o sensors exist [ 92 ], which could be combined with a red-shifted genetically encoded voltage indicator [ 93 ] to simultaneously monitor [K + ] o dynamics and synaptic synapse V m tuning in vivo. We hope that experimentalists in the future will use these advanced techniques to probe the existence of local dendritic [K + ] o changes, as well as their functions and the cellular mechanisms regulating them, such as astrocyte-mediated K + uptake [ 94 ].

Regarding the functional role of the [K + ] o -dependent dendritic integration modulation, we showed that [K + ] o increases can act as a volume knob and cause multiplicative gain modulation of the neuronal input–output function by boosting the effectiveness of orientation-tuned synaptic inputs ( Fig 4 ). This makes the neuron more responsive and could contribute to increasing the signal-to-noise ratio of visual cortical neurons while, importantly, maintaining their orientation selectivity. Compared to long-term plasticity mechanisms, [K + ] o -dependent dendritic integration modulation requires minimum usage of resources and is transient, operating in time scales of seconds, selectively boosting the activity of repetitively active dendrites; as such, [K + ] o changes closely follow the overall dendritic activity levels. Indeed, we have previously shown that [K + ] o changes are state dependent, e.g., on whether the animal is in quiescence or locomoting [ 14 ], indicating that state-dependent modulation of sensory processing may be supported by [K + ] o changes following the overall activity levels. In addition, the broadening of dendritic spikes could potentially also enhance the capacity of the neuron to integrate temporally delayed excitatory inputs [ 86 ] and act as a cellular mechanism involved in short-term memory. Notably, the [K + ] o —dependent increase in the reliability of dendritic spikes is of potential high functional importance as dendritic spikes are critical for in vivo learning and behavior [ 87 ]. For example, the dendritic response to inputs arriving on the apical dendrites of cortical neurons has been implicated in contextual modulation of sensory processing and perceptual sensitivity [ 88 – 90 ]. A recent report suggests that densely localized thalamic inputs to apical dendrites might regulate such modulation by facilitating dendritic spikes [ 91 ]. It is thus tempting to speculate that local activity-dependent [K + ] o increases of spatially clustered similarly tuned inputs to apical dendrites effectively modulate cortical sensory processing, by means of promoting dendritic spikes. In this context, it is important to note that while we chose here to use visual orientation tuning as our framework of study, our proposed mechanistic concept is agnostic to sensory features, brain regions, or animal species. The fundamental requirement for the emergence of dendritic [K + ] o hotspots is that synaptic inputs are activated on a spatially and temporally synchronized scale. Hence, this mechanism could play a role in dendritic computations in diverse brain regions such as the hippocampus, motor cortex, visual cortex, and somatosensory cortex where spatiotemporally synchronized inputs have been observed [ 32 , 42 , 45 – 50 ].

Dendritic processing of synaptic inputs depends on the spatial and temporal organization of the inputs: spatially dispersed and asynchronous inputs are summed linearly, while inputs that are spatially localized and synchronous are summed nonlinearly and can facilitate the generation of dendritic spikes [ 32 – 38 , 82 , 83 ]. Here, we propose the novel idea that another important function of grouping co-tuned synaptic inputs close in space is to generate higher activity-dependent [K + ] o increases, creating local and feature-tuned dendritic “[K + ] o hotspots.” Such [K + ] o hotspots, potentially attaining up to 5 mM K + increases relative to baseline ( Fig 1 ), can markedly affect dendritic processing by dampening K + currents. For example, we show here that they can reduce the amount of excitatory drive needed to trigger dendritic spiking, as well as broaden the dendritic spikes ( Figs 2 and 3 ). Our results on dendritic integration are supported by previous work showing that K + channels, highly expressed in dendrites [ 84 ], can modulate dendritic properties, albeit using artificial pharmacological or genetic disruption of dendritic K + channels [ 43 , 44 , 84 , 85 ], or global [K + ] o elevations [ 82 ]. Here, we introduce the novel concept of activity-dependent, local [K + ] o changes that increase the excitability of dendrites and prolong dendritic spikes.

We have developed mathematical formulations and biophysical models to address the open question of how local activity-dependent changes in [K + ] o affect dendritic integration of sensory-tuned synaptic inputs. Our work provides three major insights into this question. First, assuming activity-dependent [K + ] o changes in dendritic segments, the fine-scale arrangement of orientation-tuned synaptic inputs determines the magnitude of these changes; that is, segments with similarly tuned inputs can attain substantially higher [K + ] o increases than segments with diverse inputs. Second, these [K + ] o elevations in turn depolarize the E K + which enhances dendritic excitability, increasing both the reliability and the duration of sensory-evoked dendritic spikes. Finally, these local dendritic effects promote gain amplification of neuronal input–output functions, resulting in increased somatic responsiveness without affecting the feature selectivity of the neuron. Our results, therefore, suggest a prominent and previously overlooked role for local activity-dependent changes in K + concentration in regulating dendritic computations, shedding new light on the mechanisms underlying sensory integration in neurons.

Methods

Sampling of orientation-tuned synapses of dendritic segments We defined a dendritic segment as a cylinder with length L = 10 μm and radius R 1 = 1 μm [32,45–47]. Each segment was randomly populated with 7 to 13 synapses, yielding synaptic densities comparable to that of neocortical neurons [51–53]. The orientation preference of individual synapses of a dendritic segment depended on the synaptic organization type to which the segment belonged, that is, the diverse or similar input tuning regime. Synapses in segments with similar tuning were assigned an orientation preference using a half-circular normal distribution with σ = 15° [42,72,108] and a mean target orientation arbitrarily set to 0°, while synapses in segments with diverse tuning were assigned orientation preferences using the uniform distribution. Importantly, for both types of input regimes, we could reproduce the synaptic orientation preference cumulative distributions measured previously in the visual cortex of ferrets [42]. Moreover, the orientation preferences of our diverse tuning regime resembled those previously recorded in the visual cortex of mice [45,54]. All synapses, irrespective of the spatial organization regime, had a tuning curve width given by a half-circular normal distribution with σ tuning = 11° [42,55], and the activity of each synapse spanned from 0 to 1 (Fig 1c). This sampling method for the number and tuning properties of excitatory synapses of dendritic segments was implemented throughout this work.

Synaptic activity factor of dendritic segments We modeled the activation of synapses of dendritic segments by assuming that all synapses were activated by visual stimulation according to their tuning curve. To assess the expected synaptic activity level of a segment for a given stimulus orientation, we calculated the average activity of all synapses in that segment, w(θ), that depends on the orientation tuning of each synapse and the distribution of orientation preferences in the dendritic segment (Fig 1d and 1e). We then obtained the synaptic activity factor by comparing the expectation values from the synaptic activity distributions for the 2 types of dendritic segments. We sampled 104 dendritic segments and used a Gaussian kernel density estimator to determine the probability density function p(x) for the expected synaptic activity distributions for each segment type. The expectation value was calculated as E(X) = ∑ i x i p i where x was a linearly spaced set of values between 0 and 1. To investigate the stimulus orientation-dependent modulation of the synaptic activity factor (Fig 1f), we repeated this procedure for all stimulus orientations in the interval θ ∈ [0: 90°].

Extracellular space surrounding the dendrite segment The exact size and shape of the extracellular space surrounding functionally mapped, short dendritic segments is currently unknown, so we chose to model it as a cylinder that encapsulates the dendritic segment cylinder. The space in between these 2 cylinders thus constitutes the extracellular space in our investigations, and its volume (V ext ) is given by: (4) With R 1 and R 2 being the radius of the inner and outer cylinder, respectively, both with length L = 10 μm. This can be rewritten to express the volume ratio (V R ) as the ratio between the 2 cylinders: (5) For simplicity, synapses were considered as points on the surface of the inner cylinder, and hence, the volume of a synapse was considered part of the dendritic segment volume.

K+ diffusion state During synaptic excitatory transmission, K+ ions are released into the extracellular space primarily through NMDA receptors [16,21,26,27] and move around due to diffusion, although their movement is hindered by various cellular elements within the extracellular space. To account for this hindrance, we define effective diffusion as a function of tortuosity [66]: (6) Here, is the diffusion of free K+ ions in physiological saline, λ is a non-dimensional measure of how hindered an ion is in its movement in a given space, and D* is the resulting diffusion rate. The parameter for and λ were based on previous work [57,66,109,110]. Here, we assume that diffusion in the radial direction is negligible, as the extracellular space is very small compared to the dendritic segment length. Moreover, we assume closed boundary conditions on the outer cylinder wall, since the extracellular space mainly contacts cellular membranes of neighboring neurons and/or glial cells. Finally, assuming the extracellular K+ behaves like a brownian particle, the spatial diffusion of the K+ will extent as a multivariate normal distribution with standard deviation [111,112]. This equates the average distance traveled for an individual particle, after a certain characteristic time τ. Using this equation, we can likewise compute the time it will take for a particle to diffuse at a distance L: (7) After this characteristic time τ, we assume that the [K+] o is well-mixed and uniform throughout the extracellular space. For the dendritic segment with length L = 10 μm, the characteristic time becomes τ ≈ 66 ms. Considering that the majority of K+ efflux occurs within ≈ 100 ms from initial synaptic activation, we predict a peak [K+] o elevation along the dendritic segment after ~150–170 ms, for a τ ≈ 66 ms. Following, pumps and channels return the K+ levels to baseline, yet operating on much lower timescales as seen in in vitro experiments [58,59] (see S2 Text and S9–S13 and S15 Figs for simulations that describe the spatial and temporal properties of K+ diffusion in the presence of Na+/K+ pumps). Thus, we analyzed our models with fixed [K+] o changes, with values chosen as described in the section below. This time scale separation, due to the different time constants of the described phenomena, has been successfully applied previously on similar questions [113]. Finally, we assume that the boundary between the dendritic segments is closed, implying a finite ion concentration within the segment. This limits our approach since we do not investigate the effect of spillover K+ ions moving along the dendrite, potentially affecting neighboring segments. However, given the average distance traveled for a K+ ion between the stimuli considered here (based on Eq 7, the expected movement is ≈ 30 μm during a period of 600 ms), only the directly neighboring segments could potentially be affected, thus keeping the ionic shift highly localized (S10 Fig).

Activity-dependent K+ change across the neuronal membrane To estimate synaptic activity-dependent [K+] o changes near dendritic segments, we assumed that local increases in [K+] o are linearly correlated with the expected synapse activity, as a result of K+ efflux from the intracellular space. By scaling the K+ change in the diversely tuned input regime with the stimulus orientation-dependent synaptic activity factor, we obtained an estimate of the K+ change in the similarly tuned regime. Using this, in conjunction with Eq 5, we derived equations for the 2 types of input regimes: (8) (9) With Δ[K+] o and Δ[K+] i being the change in extracellular or intracellular [K+], respectively, V R the volume ratio as defined in 5 and E(θ) being the stimulus orientation-dependent synaptic activity factor. Using the Nernst equation, we then can express the shift in E K + (see Eq 2).

Point-dendrite model We simulated the dendritic segment V m as an isolated resistance-capacitance circuit, similar to the common practice for point-neuron models [70]. The major differences between point-neuron models and the point-dendrite model developed here stem from the specific ion channel setup and cellular resistance, chosen to mimic dendritic properties [73] (S1 Table). The circuit is given by the ordinary differential equation: (10) C m Is the membrane capacitance, set to 2 to simulate the contribution from the spines [114], I ext is the current generated by synaptic receptors (AMPA and NMDA receptors), and I int is the current generated by intrinsic ion channels (K leak , Na V , K V , K M , K A , K Ca , C aV , and HCN). The latter is multiplied by the surface area, A, as these channels are scattered uniformly along the surface of the dendrite. All relevant channel dynamics and specific parameters are included in the S1 Text. The system was simulated using an exponential Euler scheme (cnexp): (11) Here, V ∞ is the solution to the equation : (12) Where denotes a channel’s average conductance, nx is from the individual channel dynamics controlling the gating and channel-specific conductance, E ion denotes the relevant reversal potential and sums the current from the different types of synaptic receptors, were . For complete synaptic equations for NMPA, NMDA, and GABA A receptors, see S1 Text. The membrane time constant is given as follows: (13) Here, g m is the total membrane conductance ( ). The point-dendrite model was populated with orientation-tuned synapses as described in the previous section. The orientation-tuned synaptic activity in the point dendrite was implemented by multiplying g AMPA and g NMDA of synapses with the average synaptic activity, w(θ), that depends on the orientation tuning of each synapse and the distribution of orientation preferences in the dendritic segment, according to Fig 1d and 1e. For each stimulation presynaptic event, all synapses were activated with a Poisson distributed delay (λ = 80 ms). The orientation tuning of the individual synapses was achieved by scaling the AMPA and NMDA conductance according to the synapse’s tuning curve given by a half-circular normal distribution with σ tuning = 11° (see Fig 1c) [42,55,72,108]. To create a stimulation train consisting of several input events, we replicated the synaptic activation 2 times with a constant delay of 300 ms in between events. Based on our analysis in the Methods section “K+ diffusion state,” the S2 Text, and the S9–S13 Figs, we modeled the activity-dependent local increase in [K+] o a step function in between events: The first stimulation event was regarded as the control condition with ΔE K + = 0 mV, and the 2 subsequent events as the experimental condition for a given E K + shift in the interval ΔE K + ∈ [6:18 mV]. We used a V m of –30 mV as a threshold criterion for identifying dendritic spikes. This value was chosen because we observed the largest change in V m around this value, as synaptic integration transitions from subthreshold input summation to suprathreshold spiking. To evaluate the effect of nonspecific inhibitory (GABA A ) input, we repeated our point-dendrite model simulations in the presence of inhibition. Specifically, we simulated the activation of GABA A synapses randomly activated at 10 Hz (S2 Fig).

Linear dendritic spike function To investigate the relationship between the average synaptic activity, w(θ), (Fig 1e), the ΔE K +, and the number of synapses (N) for the emergence of a dendritic spike, we independently varied each variable in small increments in the point-dendrite model. By doing so, we identified the lowest value of ΔE K + that was able to push the V m over the threshold of –30 mV, our working criterion for dendritic spike generation, given N and w. We assumed that the relation between the 3 parameters was linear, and we, therefore, fitted a plane to obtain a model of ΔE K + needed to trigger dendritic spikes as a function of N and w. For this, we used the planar equation αw + βN + ν = 0 and solved it as Ax = b (S3 and S4 Figs and Table 1 for coefficients): (14) PPT PowerPoint slide

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TIFF original image Download: Table 1. Fitted coefficients for dendritic spike emergence seen in Fitted coefficients for dendritic spike emergence seen in S3 Fig https://doi.org/10.1371/journal.pbio.3002935.t001 From this equation, we can see that adding 1 additional synapse would lower the required ΔE K + for a dendritic spike by 2.08 mV. Accordingly, a 10% increase in w would lower the required ΔE K + for a dendritic spike by 3.8 mV. Note that in this work w varies between 0 and 1 and the maximum number of synapses on a dendritic segment is 13 [51–53]. A combination of sufficiently high number of synapses N and w would lead to negative ΔE K +, in which case we assumed to correspond to no E K + shift. From this, we could then estimate the dendritic spike probability as a function of ΔE K + and stimulus orientation. Using statistical formalism for the setup of the dendritic segments with similarly tuned inputs, we sampled the number of synapses and synaptic orientation preferences of 104 segments, and calculated the average synaptic activity, w(θ). Following, we determined the ΔE K + needed to generate a dendritic spike according to ΔE K + (N, w) = αN + βw + ν for each stimulus orientation. For a stimulus presented in the target orientation, we compared this value with a set of ΔE K + values in [0, 6, 10, 18 mV] to determine whether each sampled segment fired a dendritic spike or not. To obtain the set ΔE K + values for the other orientations, we multiplied the set ΔE K + values of the target orientation with the normalized version of the synaptic activity factor seen in Fig 1f and repeated the comparison with the respective needed ΔE K + value. Finally, we calculated the fraction of dendritic segments, at each orientation, that would be expected to show dendritic spikes and converted this into a probability.

Current-voltage attractor landscape To understand the biophysical principles governing the regulation of dendritic spikes by local K+ changes, we used dynamical systems theory [74]. While our point-dendrite model is defined by the differential Eq 10, it is not possible to solve it analytically. Rather, we chose to plot the exact solutions to the differential equations evaluated at different V m as a vector plot, which in essence is similar to summing all I-V curves from the included intrinsic and synaptic channels. We thus have V m on the first axis, and on the second we have its derivative which in our case is denoted . The time evolution of the synaptic currents in the point dendrite model was described as before (see S1 Text). Synaptic receptors were simulated in the time interval [0: 100 ms] using the Euler method. For each time step, an I-V curve was generated and the resulting I-V curves were concatenated to create a time-dependent I-V landscape (Fig 3e and S1 Video). To compare the effect of [K+] o changes, we simulated the system with ΔE K + = 0 mV and ΔE K + = 18 mV.

Abstract neuron model Using the NEURON simulation environment, we constructed an abstract neuron model to systematically test the role of local [K+] o elevations on dendritic input integration and somatic output firing. The dendrite’s morphology was constructed as a fractal tree (Fig 4a): for each dendritic section generation, 3 new sections, each with half the length of the former, were added (length of the most distant section: 20 μm). This gave a fractal with dimension D = ln 3/ ln 2 = 1.58, mimicking the generic compartmentalization and fractal dimensionality of the apical tree of pyramidal neurons [75,76]. The fractal dendritic tree was extended with a trunk dendritic compartment (500 μm) and a soma (length 30 μm, diameter 10 μm) to complete the neuron morphology, and intrinsic ion channels (K leak , Na P , Na T , K P , K T , K DR , K Ca , KM, Ca LVA , Ca HVA , and HCN) were inserted into the cellular compartments (S2 Table), based on [73]. Using this morphology enabled us to test the effect without hidden edge cases caused by morphology, as regardless of the placement of the cluster the effect should remain the same. The fractal dimension kept the properties of the neuron as close as possible to that of a detailed morphology. To investigate the compartmentalization of dendritic integration under the different ΔE K + values, we stimulated synapses residing in segments with similar orientation tuning on the most distant dendritic branches [38,77,78]. For this, a Poisson-distributed number of dendritic segments (λ = 32 segments) were selected to host the similarly tuned inputs in each simulation iteration, and the number of synapses and their orientation tuning were sampled as described above. Only distal dendrites received direct input, to ensure a constant distance to the soma. The local ΔE K + of stimulated dendrites were grouped into 3 conditions: no shift (ΔE K + = 0 mV), small shifts (ΔE K + = 6 mV), or large shifts (ΔE K + = 18 mV), based on the [K+] o changes found in Fig 1. As before, the orientation tuning of individual synapses was achieved by scaling the AMPA and NMDA conductance. To simulate a realistic somatic firing pattern, the synapses within each dendritic segment were activated randomly given by a Poisson distribution with λ = 30 ms. This stimulation protocol was repeated 3 times with a 300 ms delay between stimulation events. Each neuron setup was simulated with each of the 3 ΔE K + conditions to directly assess the effect of the local K+ changes.

Multiple ionic contributions In this investigation, we only considered the activity-induced changes of [K+] o and assumed the concentration of all other ions to remain constant. We assume contributions from Na+ and Ca2+ to be minimal and their effects covered within the range of the E K + demonstrated. This difference in the respective contributions arises from the ratio between the intra and extracellular volume, where the former is assumed to be at least a factor of 2 larger [62–64] (the size of the extracellular space ranges from 0.04 to 0.5 μm [63,104]), and is smaller than the pyramidal neuron dendritic diameter [105,106] and is also motivated by our previous experimental recordings documenting in vivo elevation of [K+] o [14]. This results in greater changes seen in the extracellular space, and combined with the formulation of the Nernst equation, the shift in E K + outweighs the other ions. For a more in-depth discussion about this, see S3 Text and S14 Fig.

Neuronal activity measures To obtain a better understanding of how the local [K+] o changes impact the activity of the neuronal model we introduced 2 additional measures, in addition to the conventional measurement of somatic firing rate: Neuronal firing gain transformation and area under the curve (AUC) for the integrated dendritic V m signal.

Neuronal firing gain transformation We assumed that the gain modulation of the somatic firing tuning curve could be described by a multiplicative or additive function, similar to previously described in the visual cortex of mice [79]. The multiplicative and additive functions are given by: (15) (16) Here, FR Before (θ) and FR After (θ) are the somatic firing rates at a given orientation with non-shifted or shifted E K + values, respectively, ξ is the fit parameter that describes the change in firing rate either as a gain coefficient or as a gain constant for the multiplicative and additive gain transformation, respectively. To determine the fit parameters, we fitted the 2 functions to the firing rate tuning curve data using the χ2 fitting method (Table 2 and S7 Fig). Only the multiplicative method was able to produce a satisfying fit and capture the main features of the tuning curve, whereas the additive only captured the responses to orientations close to the target orientation. PPT PowerPoint slide

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TIFF original image Download: Table 2. Fitted gain parameters. Gain parameters obtained for the multiplicative (top) and additive functions (bottom) for the small and large E K + shifts (left and right, respectively). Errors were assumed Gaussian and reported as standard deviations. See also S7 Fig. https://doi.org/10.1371/journal.pbio.3002935.t002

Area under the curve To understand how dendritic E K + shifts shape dendritic gain, we computed the AUC of the V m measured at the base of the dendritic trunk, over an interval of 500 ms to capture the full synaptic activation during an event. To exclude the contribution of bAPs, we here turned off the voltage-gated sodium channels in the soma and trunk, which eliminated somatic firing. To calculate the AUC, we first subtracted the V m baseline of the control signal, corresponding to ΔE K + = 0 mV, from all traces, and used the trapezoidal method.

Orientation selectivity index The orientation selectivity index was computed as 1 –circular variance (CV) in orientation space given by: (17) Where r k is a measurable output at orientation θ k (in radians); here, we used dendritic spiking probability and somatic firing rate as output measures. To compute OSI, we mirrored the results obtained in the simulation between [0:90°] to obtain values in the interval [−90:90°].

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