Step and hold (SH) protocol.

The Step and Hold protocol was set up in a current clamp configuration, where the resting membrane potential was set to -70 mV before current injection into the soma of the neuron. In total, 10 current step injections, each 500 ms long, were performed. The current steps were fixed and were depolarizing in nature, ranged from 40 pA to 400 pA with an inter-sweep interval of 6.5s. The stimulus was repeated 1 to 3 times for neurons. The drifts encountered were not corrected for.

Frozen Noise (FN) protocol.

The Frozen Noise protocol involves injecting a somatic current generated by a simulated Poisson neural network of 1000 neurons. Each neuron fires Poisson spike trains modulated by a binary hidden state, which represents the presence or absence of a stimulus and switches according to a Markov process. The hidden states have a time constant of either 250 ms or 50 ms. The input generated with a time constant 250 ms is used for neurons that don’t respond to a rapidly fluctuating input, these are suspected to be excitatory in nature. The spike trains are convolved with an exponential kernel (decay time 5 ms) and linearly combined with weights where and are the neuron’s firing rates in the on and off states, respectively. The resulting current is scaled and injected into the neuron, and the membrane potential is recorded at 20 kHz for 360 seconds. During the FN recording process, neurons were initally observed under a microscope and then were probed with an initial step input. Based on the response, the FN recordings were performed either with 250 ms FN input when neurons showed a slow firing response (<50 Hz) to the probe input along with a triangular shaped soma, and with a 50 ms input otherwise [19]. The protocol is repeated for control (aCSF) and drug (agonist) conditions, and the data are saved as MATLAB structs with metadata including cell type, baseline, scaling factor, and experimental parameters. See [17,18] for further details.

Feature extraction.

Feature extraction is performed separately for SH and FN protocols to classify neurons using different inputs and to determine cell class with a realistic stimulus. For the first part, which is comparing classification across protocols, we collect waveforms and Action potential attributes for comparing SH and FN protocols using a subset of the data (186 neurons). As not all cells were recorded with both types of input protocols, this subset was chosen to match the cell IDs across the two protocols. For the second part, which compares different physiological attributes for the FN protocol, we collect waveforms, action potential features, biophysical features, and the STA for each control trial for each cell in the FN dataset. We discard trials that contain recording artifacts (observed distortions or high-frequency noise), in total 11 neurons. Since there were multiple drug and control trials available for each neuron, we always took the first control trial for this study as this prevented any residual effect from the drug condition after washing. The total number of neurons used for this part of the study was 312.

Spike waveforms. Spike waveforms were extracted for each neuron from both SH and FN protocol trials (this could not be done for all the neurons as some neuron IDs could not be matched between SH and FN protocols due to missing metadata). Firstly, we identified the spike times in each trial in each protocol. This was done by identifying peaks in the membrane potential trace using the Scipy Findpeaks function [25] with and distance = 80ms as the chosen hyperparameters. Next, we cut the spike waveform by defining a window of 2 ms before and 3 ms after each spike peak to get a 5 ms long waveform for each spike. We ignored the spikes that had an ISI lower than 3 ms as we were interested in non-bursting type spike shapes. We then average over all the waveforms for each trial to get an average waveform shape in both FN and SH protocols respectively.

Extracted Waveforms for the second part were 10 ms long (5 ms before and after the spike peak), to incorporate the subthreshold dynamic before reaching the threshold of the neuron in the waveform classification as well. We observed some variability in the baseline membrane potential values as well as the slope of membrane potential before the threshold value was reached.

Action potential features. Action potential features were extracted to study the variability in spike-related dynamics across individual neurons. These features were designed based on their suitability for comparing neurons in FN and SH protocols. These Action potential features were divided into three categories: spiking dynamics, spike threshold, action potential height and width.

Spiking dynamics.

This includes the following features:

• Current at the first spike: We take the current amplitude when the membrane potential crosses the threshold for the first time in the trial for FN protocol. For SH protocol, it is the current step that produces the spike for the first time.
• AP count: We count the number of spikes over the entire trial length of 360 seconds for the FN trial and 500 milliseconds for the SH trial which is the duration for the current onset.
• Time to first spike: We measure the time (in milliseconds) it takes for the neuron to fire the first action potential. For the SH case, we take the lowest amplitude step where a spike is observed.
• Firing rate: We calculate the firing rate as the number of spikes per second. For the FN case, we take the entire length of 360 seconds of the trial and for the SH case, we take the duration of the current onset which was 500 milliseconds for the highest current step (400 pA). (1)where N_spikes is the total number of spikes in the trial and T is the total length of the trial.
• Interspike Interval: Interspike interval is the duration between two spikes , where and are spike times for spikes n+1 and n. We take the interspike interval for all the spikes in the trial for both FN and SH protocols and calculate the mean, median, minimum, and maximum values.
• Instantaneous rate: Instantaneous rate is defined as the reciprocal of the average of the Interspike interval. (2)

Biophysical feature extraction using GLIF model.

Since it is not possible to empirically observe the passive biophysical properties of the cell just using the membrane potential, we fit a GLIF model to the recordings, which can capture universal spiking and sub-threshold dynamics [27]. The following equations define the GLIF model:

(3)

where V(t) is the membrane potential, C is the membrane capacitance, gL is the leak conductance, EL is the resting potential, and is the adaptation current triggered by a spike event. Spikes are stochastically produced by a point process that represents conditional firing intensity that is dependent on the instantaneous difference between the membrane potential and voltage threshold given by:

(4)

where is the base firing rate in Hz, V(t) is the membrane potential, and (t) is the moving spike threshold and controls the sharpness of the exponential threshold function. The probability of a spike between a time interval t and is given by the following equation (based on [28]):

(5)

The dynamics of the firing threshold are given by:

(6)

where is the stereotypical movement of the spike threshold after a spike and is the threshold baseline.

The method for fitting this neuron model to a membrane potential recording is divided into the following steps:

1. Preparation step: A 100-second window from the initial part of the trial is taken as the training set for the fitting, [27] shows that a longer trial length doesn’t improve the fit. Spike times and waveforms are also extracted for the preparation step of the fitting procedure.
2. Step 1. Fitting the reset voltage: The waveforms extracted in the preparation step were averaged, and then the Reset voltage was extracted using the averaged waveform by setting an arbitrary refractory period t_ref and taking the membrane potential value at , where is the spike peak. The refractory period is chosen to be always lower than the minimum inter-spike interval, we chose the refractory period of t_ref = 4ms in this case.
3. Step 2. Fitting sub-threshold dynamics: The voltage dynamics in Eq (3) are given by parameter set ={C, g_L, , and E_L}, by fitting the temporal derivative of the data in the model, we can extract the set of passive parameter set for the data. Firstly, we can write the adaptation current as a linear sum of rectangular basis functions [29].
   (7)
   Using the fact that the voltage dynamics are approximately linear in the subthreshold regime, parameter set can be extracted using a multi-linear regression between and . For this, we created a training set where we removed the spike waveforms from , , where is the spike times. The regression problem can be stated as:
   (8)
   where X^T is a matrix representing parameter values at different time points, its row elements are of the following form
   (9)
4. Step 3. Fitting the spike probability: For fitting the spiking probability to the data, we need to extract parameters defining the dynamics of the threshold Eq (6). The stereotypical shape of the adaptation current threshold movement can be expanded as a sum of rectangular basis functions as follows [29]:
   (10)

We use the parameters obtained in the previous steps to compute the subthreshold membrane potential of the model using numerical integration of Eq (3). We set Hz and all the threshold parameters , , and } are extracted by maximizing the likelihood function of the following form based on the experimental spike train:

(11)

Where = {t|t (,

The subthreshold fit is examined by comparing the variance explained R² of the subthreshold membrane potential trace V between the data and the model. All the models chosen for clustering had an R² value > 0.7. The sets = {g_L, , C, , E_L, } are the parameters that are extracted from the model that is used in the clustering procedures.

Spike triggered average.

The STA is the average shape of the stimulus that precedes each spike. We extracted the STA using the following equation given by [30]:

(12)

where t_n is the n^th spike time, s is the stimulus vector preceding the spike for a fixed time window of 100 ms, and N is the total number of spikes. Before clustering, we standardize (i.e. z score) and then normalize the STA vector with an L₂ norm. We didn’t use any kind of whitening or regularization to calculate the STA.

UMAP.

Universal Manifold Approximator (UMAP) is a non-linear dimensionality reduction technique that preserves local and global relationships between data in high dimensional space [32]. It is divided into two steps, the first step is creating a k-nearest neighbor graph and the second step is to generate a low-dimensional representation that is similar to the high-dimensional graph structure.

We used the Scikit-learn UMAP-learn software package [36] to extract the embedding and the graphs.

Louvain community detection.

The Louvain community detection algorithm [33] maximizes modularity among the identified groups in a graph. Modularity can be defined by the following equation:

(13)

where A_i,j is the adjacency matrix, k_i = is the sum of the weights of the edges attached to the vertex i, c_i is the community for vertex i, d_i is the degree of node i, and are the in degree and out degree for node i,, Kronecker symbol, is 1 if u=v and 0 otherwise, w = A_i,j and is the resolution parameter. For Louvain graph-based clustering we used the implementation from the Scikit-network software package [37].

The clustering approach can be summarized in the following steps:

1. The high dimensional k-neighbor graph is obtained by the first step of the UMAP algorithm using data vectors that are first standardized and then normalized using the L₂ norm. The nearest neighbor and distance parameters for UMAP were 20 and 0.1 respectively. This is to ensure a compact embedding and a clear clustering.
2. Using the graph obtained in the first step, we perform Lovain community-based clustering, using the resolution parameter that maximizes the modularity score, and the corresponding community/partition is chosen as the final cluster labels.
3. Using the cluster labels found in the second step, we color the individual points in distinct colors on the low-dimensional UMAP representation.

This unsupervised clustering approach was effective in capturing the global structure of the high dimensional space across attributes such as waveforms, adaptation current, and Spike Triggered average. The clusters found using this method were robust even when clustering was repeated with a sub-sample of the data and while iteratively removing the features.

Cluster stability and parameter selection. Cluster stability is tested by clustering a 90% sub-sample of the data chosen at random and repeating the procedure 25 times for each resolution parameter, varying from 0 to 5 with a step of 0.5. The modularity score is calculated for each resolution parameter and finally, the resolution parameter is chosen for which the modularity score is maximal. Variation in the number of clusters is observed for each resolution parameter and is contrasted with the modularity score. Clustering robustness is also tested by repeating the procedure above while excluding one feature at a time for Action potential feature clustering and passive biophysical clustering.

Cluster likelihood comparison. Cluster likelihood between two sets of labels was calculated in two steps, firstly we created a contingency matrix such that C_i,j contained the number of times neurons classified in cluster i in the SH protocol classified as cluster j in the FN protocol, such that each row contains the division of elements of cluster i in the SH protocol into all the clusters of the FN protocol. Secondly, to get the likelihood, we divided each row by the total count of neurons in cluster i in the SH protocol.

(14)

Here is the probability of the neuron classifying into FN cluster j given it is classified in SH cluster i.

Cluster similarity measures. We measured the similarity between two given cluster assignments using the Adjusted Random Index (ARI) and Adjusted Mutual Information (AMI) Score. Both of these measures amount to change agreements between two clusters. The ARI (range 1-1) measures the pairwise relationships between clusters and the AMI (range 0-1) measures the overall information shared between the two clusters.

We used the scikit-learn Python package [38] to calculate the ARI and AMI measures.

Ensemble Clustering for Graphs (ECG).

We used Ensemble Clustering for Graph algorithm [34] to validate the clusters found using the Louvain Community detection algorithm. It is a two-step algorithm, the first step called the generation step consists of producing a k–level 1 partition by running the first pass of the Louvain clustering algorithm with random vertices on the initial graph G= (V, E). The second step, also known as the integration step, consists of running the Louvain algorithm on a weighted version of the initial graph. Where the weights are the weight of an edge given by

(15)

where 0 < w_* < 1 is the minimum ECG weight and shows if the vertices u and v co-cluster in the same cluster of P_i or not. Thus it takes advantage of multiple instances of the Louvain clustering algorithm to make a clustering based on consensus. We used the implementation provided by [35] for comparing the waveform clustering for SH and FN protocols. The graph used for Ensemble clustering was the same as in the original clustering using the UMAP algorithm.

Stimulus dependence of neural classification using intracellular action potential waveforms and attributes.

To understand whether waveform as well as action potential attribute based classification differs under FN and SH input conditions, we analyzed control (aCSF) trials from a total of 186 in-vitro whole-cell patch-clamp recordings, where the same cell was recorded under two different input protocols: SH and FN. It has been shown that excitability measures such as total spike count and AMPA conductance threshold (dynamic) versus rheobase (static) have a low correlation between static and dynamic stimulus conditions [42,43]. This motivated us to examine whether the classification of neurons into functional types depends on the stimulation protocol used. For waveform-based classification, we extracted waveforms of equal length (5 ms) from both FN and SH trials, standardized them (subtracting the mean and scaling to unit variance), and then normalized the data using an L2 norm. Similarly, for action potential attribute-based classification, we extracted attributes that incorporate, among others, spiking dynamics, spike threshold, and action potential height and width; a total of 22 attributes (see Methods). This attribute set was designed to enable comparison across input protocols. We used descriptive statistics such as mean, median, minimum, and maximum values for some of the features to capture their distribution within each trial. The data were then standardized and normalized as done with the waveforms.

Next, we apply an unsupervised high dimensional clustering algorithm that combines UMAP and Louvain Community detection (see Methods) and found 7 clusters for SH trials as well as 7 clusters for FN trials (Fig 1A and 1B) respectively. A sample trace from FN and SH trials are shown in Fig 1C and 1D. Applying the same clustering method on the 22 action potential attribute set, we found 7 clusters for SH and FN trials respectively (Fig 2A and 2B). The AP attributes for each trial (thin line) along with the mean for each class (thick line) is shown Fig 2D and 2E. On visual inspection, we observe that the means for each cluster (thick line) in both SH and FN protocols are non-overlapping for all the 3 attribute sets (spiking dynamics, spike threshold, and AP height and width) respectively. Suggesting that each cluster has distinct action potential attributes. We also measured the stability of the clusters (see Methods) against changing the hyperparameter (resolution parameter) for the unsupervised method. We found the clusters to be stable (low standard deviation) for the chosen resolution parameter (chosen at a value of 1.0) (See Fig 1G and 2C). A 5 second long sample trace of each FN waveform class along with it’s corresponding SH trial is shown in S7 Fig.

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Fig 1. Intracellular waveform-based neural identity depends on the stimulus type.

Each of 186 cortical neurons was recorded under two stimulation protocols: Step-and-Hold (SH) and Frozen Noise (FN). (A-B) UMAP projections of spike waveforms from FN and SH protocols, colored by cluster identity (top) with their corresponding average waveform shapes (bottom). (C-D) Representative 200 ms traces from FN and SH recordings showing broad- (red) and narrow-width (blue) spike waveforms. (E) Within-protocol cosine similarity matrix showing inter-class waveform similarity for SH and FN clusters. (F) Cross-protocol cosine similarity matrix comparing waveform shapes between SH and FN clusters. (G) Cluster stability analysis showing mean SD modularity (left axis) and number of clusters (right axis) across resolution parameters; the black arrow marks the chosen value (1.0). (H) Likelihood of SH class membership mapping to FN classes. Low ARI (0.085) and AMI (0.133) scores indicate inconsistent class correspondence between protocols.

https://doi.org/10.1371/journal.pcbi.1013821.g001

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Fig 2. Neural identity based on action potential (AP) attributes depends on stimulus type.

(A-B) UMAP projections of AP attributes for neurons recorded under FN and SH protocols, colored by cluster identity. (C) Cluster stability analysis showing mean SD modularity (left axis) and number of clusters (right axis) across resolution parameters (0-5, step 0.5), based on 25 iterations using 80% random subsampling. The black arrow marks the chosen resolution maximizing modularity. (D-E) Radar plots showing AP attributes grouped by spiking dynamics, spike threshold, and AP height/width features for SH and FN protocols. Thin lines represent individual trials; thick lines show cluster means. (F) Cosine-similarity heatmap comparing AP attributes between SH and FN clusters, indicating no high-similarity pairs (>0.8). (G) Likelihood heatmap showing SH-FN cluster correspondence. Low ARI (0.149) and AMI (0.206) scores indicate inconsistent class identity across input protocols.

https://doi.org/10.1371/journal.pcbi.1013821.g002

We compared waveform shapes as well as action potential attribute sets across clusters by creating pairwise cosine similarity matrices between FN and SH protocols and then taking the average of the sub-matrix for each SH and FN cluster pair. The average similarity across waveform clusters is summarized using a heatmap (Fig 1E and 1F), and a similar heatmap is drawn for action potential attributes (Fig 2F). We first performed this comparison for each protocol separately and found that some cluster waveform shapes were similar (cosine similarity >0.9) to their immediate neighbors (e.g., clusters 7, 3, and 4 in SH and clusters 6, 3, and 4 in FN; Fig 1E). Comparing waveform shapes between SH and FN, we found that narrow-width clusters for the SH protocol (4, 3, and 7) were highly similar to narrow-width clusters for the FN protocol (6, 2, and 3). For the action potential attribute sets, none of the SH clusters showed a high similarity (>0.9) with FN clusters, suggesting that action potential attributes differ substantially as a result of the input protocol (Fig 2F).

We projected all the SH and FN waveforms together onto a single embedding space and observed that they were distinct (S1A Fig), suggesting that waveform shapes are not fixed across protocols. We then tested for potential drift in the recordings by dividing each 360-second trial into two parts and comparing the waveforms between the first and second halves by projecting them onto the same 2D UMAP space. The two halves overlapped, suggesting that waveform shapes remained consistent throughout the trial. We also overlaid the UMAP embedding for each action potential attribute set for the SH and FN protocols (S1C Fig) and found that the SH and FN feature manifolds were entirely different.

Finally, we calculated the likelihood that neurons in one of the SH waveform-based clusters would cluster together in one of the FN clusters (see Methods and Fig 1H). For each SH waveform cluster, the likelihood of clustering in one of the FN clusters was broadly distributed without a strong majority, suggesting that neurons grouping together under the SH protocol do not group similarly under the FN protocol. We further quantified the clustering agreement between the SH and FN clusters using the adjusted Rand index (ARI) and adjusted mutual information (AMI). Both measures were low (ARI = 0.085, AMI = 0.133), confirming that neurons based on their waveform cluster differently between SH and FN protocols. For action potential attribute sets, we found a similar pattern: the likelihood of SH cluster membership corresponding to a specific FN cluster was broadly distributed. The corresponding scores were also low (ARI = 0.149, AMI = 0.206). To verify that the found clusters were not biased by the method used, we repeated the waveform clustering analysis using the ensemble clustering method (S2 Fig). Taking the average of the highest values for each row in the co-clustering matrix, we found 84% correspondence for FN waveforms and 96% for SH waveforms between the wave map and ensemble clustering. Comparing the ensemble clustering with Louvain clustering using the adjusted mutual information score (, ), we found a high level of similarity for both SH and FN protocols.

In conclusion, clustering neurons into cell classes based on their waveforms results in different classifications under SH and FN input protocols, reflecting differences in waveform shapes due to the stimulation protocol. This shows that waveform-based neuronal identity is stimulus-dependent. Similarly, neurons cluster differently between SH and FN protocols based on action potential attributes, suggesting that action potential attribute based clustering of neuronal populations is also input-dependent.

Action potential attributes based neuron profiles using the FN protocol.

As pointed out in the previous section, we aimed to understand and compare the usefulness of commonly used physiological attributes in uncovering functional classification in neuronal populations when neurons receive a physiologically realistic FN input. For that aim, we study commonly used action potential attributes for excitatory and inhibitory populations separately to discern if action potential attributes sufficiently capture within-population heterogeneity in excitatory and inhibitory groups. We also aim to unravel the differences between excitatory and inhibitory populations regarding their action potential attributes. For this, we extract 22 attributes (see Methods) subdivided into spiking dynamics, Spike threshold, and action potential height and width attributes with their descriptive statistics incorporating the mean, median, minimum, and maximum values.

We cluster the E/I populations separately based on the features and find 7 clusters for excitatory cells (shaded in red) and 6 clusters for inhibitory cells (shaded in blue). The UMAP representation of the spike-based properties for E/I populations with unique colors for each cluster for the chosen cluster parameter is shown in (Fig 4A and 4B). We show the cluster stability (see Methods) in (Fig 4C), the number of clusters is stable (low standard deviation) for the chosen resolution parameter (black arrow). We also test the stability of the clusters by excluding one attribute at a time and repeating the stability analysis (see Methods) and find the inhibitory clusters to be more stable to attribute exclusion (S3A and S3B Fig). We compare the means of action potential attributes simultaneously between excitatory and inhibitory populations and find them to be significantly different (one-sided MANOVA (see Methods), Wilks’ lambda; ; p = 0.000, Pillai’s trace; ; p = 0.000, Hotelling-Lawley trace; ; p = 0.000, Roy’s greatest root; ; p = 0.000). To identify the relative importance of each attribute in the separation of excitatory and inhibitory populations, we perform a canonical correlation analysis (CCA) (see Methods) between the excitatory and inhibitory action potential attributes and calculate the loading (structure correlation) for each attribute (Fig 4D). We find that spiking dynamics attributes have the most influence on the inhibitory canonical variates suggesting that these variables have the most influence on separating the inhibitory population apart from the excitatory population. Alternatively, AP height and width attributes strongly influence the excitatory population canonical variate, suggesting that AP height and width are the most discriminatory. These results also show that individual action potential attributes contribute differently to the canonical variate (i.e., the latent structure) for excitatory and inhibitory populations respectively.

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Fig 4. Cell type-specific heterogeneity and the principal discriminatory attributes of the action potential.

Putative excitatory (red background) and inhibitory (blue background) neurons were analyzed separately. (A-B) UMAP-based clustering grouped excitatory and inhibitory populations into 7 and 6 clusters, respectively. (C) Chosen resolution parameters corresponding to maximal modularity and resulting cluster counts. (D) Multivariate analysis confirmed significant differences in AP attributes between excitatory and inhibitory populations. (one-sided MANOVA; Wilks’ lambda, Pillai’s trace, Hotelling-Lawley trace, and Roy’s greatest root: , ***p < 0.001). Post hoc canonical correlation analysis (CCA) shows attribute loadings with 95% confidence intervals (grey lines). (E-F) Radar plots of AP attributes grouped by spiking dynamics, spike threshold, and AP height/width. Thick lines show cluster means; thin lines represent individual neurons. Spiking-dynamics parameters exhibit the largest variability across clusters. (G) Cosine similarity between excitatory-excitatory (E-E), inhibitory-inhibitory (I-I), and cross-type (I-E) feature vectors. Inhibitory neurons display greater heterogeneity in AP attributes (Welch’s ANOVA: , p < 0.001; Games-Howell post hoc: all pairwise comparisons **p = 0.001).

https://doi.org/10.1371/journal.pcbi.1013821.g004

Next, we want to compare the level of heterogeneity within excitatory (E-E) and inhibitory (I-I) populations as well as across (I-E) populations using the action potential attributes. We visualize the action potential attributes using a radar plot shown in (Fig 4E and 4F). It shows the diversity of spiking dynamics, spike threshold, and AP height and width for each neuron in each cluster (with the same color as in the UMAP projection (Fig 4A and 4B)) along with the mean for the cluster (thick line). We observe that spiking dynamics attributes differ between excitatory and inhibitory clusters. The spiking threshold profiles were also found to be different across E/I clusters. Comparing the AP height and width attributes, however, show a similar profile across E/I clusters. We quantify the heterogeneity in the E/I clusters using a cosine similarity measure. The excitatory population had a significantly higher cosine similarity measure within the population than the inhibitory population (Welch’s ANOVA, F(2,49135)=3626998.65, p= 0.0; Post-hoc Games-Howell test, E-E vs I-I (p=0.001), I-I vs I-E (p=0.001), and E-E vs I-E (p=0.001), (Fig 4G)), suggesting that inhibitory action potential attributes are more heterogeneous than their excitatory counterparts. The mean cosine similarity score between excitatory and inhibitory populations are significantly lower than the within-population cosine similarity score for both excitatory and inhibitory populations respectively, suggesting that excitatory and inhibitory feature action potential attribute vectors are different from each other, reiterating the MANOVA results. The results above demonstrate that action potential attributes are different between excitatory and inhibitory populations. The inhibitory population is more heterogeneous in its action potential attributes than the excitatory population.

Passive biophysical feature and adaptation-based profile using the FN protocol.

As we describe in the last two sections, we aim to compare neuronal attribute sets to find the attributes(s) most informative about heterogeneity in excitatory and inhibitory populations and try to understand how neurons cluster based on these properties. The passive biophysical attributes (i.e., membrane resistance, capacitance, etc.) shape the response properties of a neuron. Moreover, it has been reported that adaptation current which captures the adaptation properties of the membrane is another important neuron property that can determine cell classes with noisy input [29]. We, therefore, consider the passive biophysical attributes and the adaptation current as potential candidates that capture heterogeneity within the neuronal population. To understand the heterogeneity of passive biophysical properties and adaptation parameters across the excitatory and inhibitory populations, we extract both a set of passive parameters (see methods) and an adaptation current by fitting a Generalized Leaky Integrate and Fire (GLIF) model on the first 100 seconds of each FN trial, using the automated method described by [27] (see methods). (Fig 5A) shows an example of a 10s instance of a GLIF-fitted model and one of the original recordings. We characterized the goodness of fit by measuring the explained variance (EV) between the subthreshold membrane potential of the original data and the model. (Fig 5B) shows the distributions of the EV and for the entire dataset. We eliminate the models with an EV value below 0.7 and obtain a set of 307 samples (out of 312 samples).

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Fig 5. Discriminatory power of passive biophysical and adaptive membrane properties for neuronal classification.

A GLIF model was fitted to FN-stimulated membrane potential recordings. (A) Example 10-s snippet showing model output (red) and experimental trace (gray). (B) Histogram of subthreshold variance explained; only models with R² > 0.7 (dashed line) were used for clustering. (C) Chosen resolution parameters yielding maximal modularity and corresponding cluster counts. (D-E) UMAP representation and radar plots of passive biophysical parameters for excitatory (7 clusters, left) and inhibitory (6 clusters, right) populations. (f-g) Same for adaptation current parameters. (H) Post hoc CCA shows attribute loadings with 95% confidence intervals. Excitatory and inhibitory passive parameters differed significantly (MANOVA; Wilks’ lambda: , ***p < 0.001; Pillai’s trace, Hotelling-Lawley trace, and Roy’s root identical). (I) Cosine similarity among E-E, I-I, and I-E passive attribute vectors showed no significant difference (Welch’s ANOVA: , p = 0.84). (J) Adaptation current similarity differed significantly across groups (Welch’s ANOVA: , **p < 0.001); Games-Howell: I-I vs E-E **p = 0.001, I-I vs I-E p = 0.75, E-E vs I-E **p = 0.001), indicating greater inhibitory heterogeneity.

https://doi.org/10.1371/journal.pcbi.1013821.g005

We cluster the recordings into cell classes based on a set of passive biophysical parameters along with threshold adaptation constants (in total 6 features, see Fig 5A and 5B) using the UMAP+Louvain clustering method explained in the previous sections, we find 7 clusters for the excitatory population and 6 clusters for the inhibitory population. We examine the stability of the clusters (Fig 5C) with different hyper-parameters (resolution parameters, see Methods) and choose (black arrow) the resolution parameter with the maximum modularity score. The clusters for the chosen hyperparameter for both excitatory and inhibitory populations were stable (low standard deviation). We also test the stability of the clusters by excluding one attribute at a time and repeating the stability analysis (see Methods) and found the inhibitory clusters to be more stable to attribute exclusion (S3C and S3D Fig). (Fig 5D and 5E) shows the UMAP representation of the feature set (inset) for excitatory (red background) and inhibitory (blue background) populations, the radar plot shows the individual passive biophysical parameter values for each neuron and each thin line in the radar plot represents a single neuron and are color matched to respective clusters, the mean for each cluster is plotted with a thick line. We compare the means of all the passive biophysical attributes simultaneously between excitatory and inhibitory populations and find them to be significantly different (one-sided MANOVA, Wilks’ lambda; ; p = 0.000, Pillai’s trace; ; p = 0.000, Hotelling-Lawley trace; ; p = 0.000, Roy’s greatest root; ; p = 0.000). To identify the relative importance of each passive biophysical attribute in the separation between excitatory and inhibitory populations, we perform a CCA analysis (see Methods) between the excitatory and inhibitory passive biophysical attributes and calculate the loading (structure correlation) for each attribute (Fig 5H). We find dissimilar contributions from excitatory and inhibitory passive parameters towards their respective canonical variate, suggesting that each passive biophysical attribute contributes differently to the excitatory-inhibitory latent structure and therefore sets the excitatory-inhibitory populations apart. Next, we compare the level of heterogeneity between the passive feature vectors across the E/I populations (Fig 5I) using the cosine similarity measure within and across excitatory populations and find no significant difference between the mean cosine similarity measure within excitatory and inhibitory populations as well as across excitatory-inhibitory population (due to low effect size) (Welch’s ANOVA, F(2,66792) = 0.16777; p= 0.84, Fig 5I). These results suggest that even though the mean of the excitatory and inhibitory passive biophysical attributes are significantly different from each other (measured using MANOVA), the level of heterogeneity within excitatory-inhibitory populations respectively based on passive biophysical parameters is quite low. Therefore the passive biophysical attributes are not so informative about the neuronal heterogeneity.

As we saw above, passive biophysical attributes are not informative about population heterogeneity, this result is consistent with [29], which finds that passive biophysical attributes do not distinguish between neuron types. The adaptation current is more useful in putting neuron types apart. Therefore we classify the neurons using the adaptation current () extracted from the fitted GLIF model for the E/I population separately. We find 6 classes for the excitatory population (red) and 5 classes for the inhibitory population (blue). In (Fig 5F and 5G) we show the UMAP representation of the adaptation current and the corresponding average normalized shapes for each cluster (inset) each cluster has its respective color in the UMAP as well as shape plots. We observe that the excitatory adaptation currents have a stronger negative amplitude than their inhibitory counterparts. Moreover, the adaptation currents of the inhibitory population relax back to their resting values at earlier times than for the excitatory population.

We quantify the heterogeneity of adaptation currents within excitatory and inhibitory populations using the cosine similarity measure. The heatmap in (Fig 5J) shows that the excitatory adaptation currents have a significantly higher average cosine similarity compared to the inhibitory adaptation currents, suggesting that the inhibitory adaptation current is more heterogeneous than the excitatory population (Welch’s ANOVA; F(2,46968) = 2796.77, p = 0.0; Post-hoc Games-Howell test E vs I (p = 0.001), I vs (p = 0.75), and E vs (p = 0.001), Fig 5J). The similarity measure is not significantly different between the inhibitory (I) population and across the inhibitory and excitatory (I vs E) population. This suggests that inhibitory adaptation currents are as different from each other as they are different from the excitatory adaptation currents. These results suggest that adaptation current profiles are different between excitatory and inhibitory populations. These results also show that adaptation currents are useful for understanding the heterogeneity within the inhibitory population but not so much for the excitatory populations.

Neuronal classification based on linear input filter approximated using a Spike Triggered Average (STA) in the FN protocol.

In last two sections, we explored neuronal heterogeneity based on action potential, passive biophysical, and adaptation attributes in neurons while responding to an FN input. We found that except for passive biophysical attributes, other properties show both within population across population heterogeneity for both excitatory and inhibitory populations. We observe a higher level of heterogeneity for inhibitory populations. Since we aim at understanding the extent to which various neuronal properties help uncover neuronal diversity, in this section we focus on the linear input filter of neurons and aim to understand the neuronal diversity based on this attribute. We consider this attribute important as neurons respond strongly to specific features in the input [49], making linear input filter an important property to study the functional diversity of the neuronal population.

The STA, as described in [30], estimates a neuron’s linear input filter by identifying the features in the input that trigger the neuron to spike. It does this by averaging the input signal over a specific time window preceding each spike, using data from all observed spikes. This makes the STA a useful method to approximate the linear input filter of a neuron. We want to investigate the STA diversity across the E/I population, for this, we extract the STA from all the neurons (see Methods). For calculating the STA, we use the injected current which is the result of shifting a theoretical dimensionless input with a constant baseline and scaling it by a factor as explained by [17,18].

We cluster the excitatory and inhibitory population STAs separately using an unsupervised UMAP+Louvain clustering method (see Methods). We find 7 clusters for the excitatory population STAs (red) and 8 for the inhibitory population STAs. The STAs were normalized and standardized before clustering (see Methods). The UMAP representation and the corresponding averaged and normalized STAs for each cluster are shown in (Fig 6A and 6B). A visual inspection shows that inhibitory and excitatory STA shapes differ in their peak amplitudes and in the maximum slope of their initial rise (Fig 6A and 6B inset), this difference is shown more clearly in (Fig 6D). The stability of the clustering algorithm for the chosen parameters is shown in (Fig 6C). To quantify the heterogeneity within the excitatory and inhibitory population as a result of that linear input filters, we first calculate the STA shape similarity between clusters, using a cosine similarity between excitatory and inhibitory STA clusters (Fig 6E and 6F). The values shown in the heatmap in Fig 6E and 6D were calculated by averaging the cosine similarity matrices between STAs of each cluster pair and within each cluster (for the within-cluster comparison, only the upper triangular values were included in the average). The average cosine similarity value between excitatory STA clusters shows higher values (>0.9) with each other except for clusters IDs 2 and 6, which show a low similarity with every other cluster except for itself. On average, each excitatory cluster shows a high similarity (>0.9) with more than 2 other clusters. On the other hand, the average cosine similarity values between inhibitory clusters is low (<0.9), with some exceptions such as cluster IDs (2, 8), (3, 6), (5, 4), (6, 5) and (8, 6). On average, the STA of each inhibitory cluster is highly similar to approximately 1 other cluster. The average cosine similarity value is high between STAs within each inhibitory cluster. We summarize the STA similarities within the excitatory (E-E) and inhibitory populations (I-I) as well as across the two populations (I-E) in the heatmap in (Fig 6G). This is done by averaging the cosine similarity matrices between STAs within excitatory (E-E) and inhibitory (I-I) populations as well as across excitatory and inhibitory (I-E) populations, these matrices are shown in (S5B Fig). For within-population averages (E-E and I-I), we only take the upper-triangular portion of the matrix. The average cosine similarity value within the inhibitory population is significantly lower than that of within the excitatory population (Fig 6G Welch’s ANOVA, F(2,48825) = 1005.90, p = 0.0; Post-hoc Games-Howell test, E vs I (p = 0.001), I vs (p = 0.001), and E vs (p = 0.001)), suggesting that inhibitory STAs are more heterogeneous than excitatory ones. Also, the STA shape similarity between excitatory and inhibitory populations is significantly lower than the within excitatory and inhibitory similarity, suggesting that excitatory and inhibitory STAs are different. We can see that on average inhibitory neurons respond to more diverse features in the input than excitatory neurons.

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Fig 6. Linear Input Filter measured using a Spike Triggered Average (STA)

(A-B) UMAP representations of excitatory (left, 7 clusters) and inhibitory (right, 8 clusters) populations with corresponding normalized average STA shapes in the 50 ms preceding a spike. (C) Resolution parameters chosen for maximal modularity and resulting cluster counts. (D) Scatter plot of maximum slope versus maximum amplitude for each STA (excitatory in red, inhibitory in blue). (E-F) Average cosine similarity within excitatory (e) and inhibitory (f) STA clusters. Excitatory clusters exhibit higher intra-group similarity (>0.9 with 2 other clusters on average), whereas inhibitory clusters show similarity with only 1 other cluster. (G) Heatmap of average cosine similarity within (E-E, I-I) and across (E-I) populations. Inhibitory neurons display greater heterogeneity (lower similarity). Welch’s ANOVA: , ***p < 0.001; Games-Howell post hoc: E vs I **p = 0.001, I vs E-I **p = 0.001, E vs E-I **p = 0.001.

https://doi.org/10.1371/journal.pcbi.1013821.g006

Comparing physiological heterogeneity across attribute sets using multi-set correlation and factor analysis.

We aim to explore what a physiologically realistic stimulus reveals about the functional heterogeneity of a neural population. For this aim, we need to compare FN-based neuronal attribute sets and find out which attribute set is the most informative about neuronal heterogeneity under the dynamic FN stimulation protocol. This is a complicated problem, as the four attribute sets are of varying dimensions. To perform this comparison, we use a method known as Multi-set Correlation and Factor Analysis [40], which is an unsupervised multi-set integration method based on probabilistic principal Component Analysis (pPCA) and Factor Analysis (FA), that can help to understand the common and shared factors across multi-modal data (see methods & [40]). We use this method to compare the private and shared variance explained by the 4 attribute sets we examined in the previous sections (i.e., action potential attributes, passive biophysical attributes, adaptation currents, and linear input filters (STA)). We use the pPCA space to model the shared structure across attributes (see Methods), and the residuals are modeled as a private structure for each attribute set using factor analysis (see methods).

Comparing the variance explained by the shared structure across attributes, we found that the excitatory population which is inferred by a 4-dimensional shared structure, explains almost 50% of the variance in action potential and passive biophysical attributes (Fig 7A top), this is higher than linear input filter (STA) (32.9%) and adaptation current attribute (19.7%). Most importantly, we see that the linear input filters (STAs) explain the highest private variance (57.2%), followed by adaptation current (41.2%). The action potential and passive biophysical attributes have a relatively lower private variance (35.7%). We can further investigate the contribution to the most important shared factor by each attribute, through which we find that action potential and passive biophysical attributes explain the most variance for the most important shared dimension (Fig 7A (middle, bottom)). Similarly, for the inhibitory population, explained by a 3 dimensional shared structure, we find that the linear input filter (STA) explains the most amount of private variance (80.6%), much higher than the excitatory population (Fig 7B (top)), followed by adaptation current (29.4%), action potential attributes (35.7%), and passive biophysical attributes (17.1%) respectively. The action potential attributes and passive biophysical attributes explain the most and almost equal amount of shared variance (33.8% and 33. 85% respectively) (Fig 7B (middle)), followed by adaptation current (29.4%) and linear input filter (9.9%) (Fig 7B (top)). We find that the adaptation current and passive features explain the largest amount of variance for the most important shared dimension (Fig 7B (bottom)) for the inhibitory population. It is important to observe that the linear input filter explains more than 85% of the total variances (shared+private) for both excitatory and inhibitory populations, higher than the other 3 attribute sets.

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Fig 7. MCFA analysis comparing shared and private variance explained by Action potential, Passive biophysical, Adaptation current, and Linear input filter (STA):

(A) Excitatory population: Histogram shows variance explained by each attribute. Action potential (AP) and passive biophysical parameters account for most shared variance (AP: shared 52.3%, private 12.7%; passive: shared 51.9%, private 2.2%), indicating low unique informativeness. STA captures most unique variance (shared 32.9%, private 57.2%) with minimal residual, while adaptation current has intermediate private variance (private 41.2%). Heatmap shows contributions of four shared dimensions, with AP and passive parameters dominating. (B) Inhibitory population: AP and passive parameters explain the most shared variance (AP: shared 33.9%, private 35.7%; passive: shared 33.85%, private 17.1%), though passive parameters also have high residual. STA explains most private variance (private 80.6%) with low residual, followed by adaptation current (private 29.4%). Heatmap of three shared dimensions shows contributions from AP, passive, and adaptation currents. Overall, STA is the strongest marker of population heterogeneity, highlighting the role of input in distinguishing excitatory and inhibitory neurons.

https://doi.org/10.1371/journal.pcbi.1013821.g007

The high values of private variance explained by the STAs in both excitatory and inhibitory cases show that passive biophysical, action potential, and adaptation currents are not highly correlated with the linear input filter of a neuron. Therefore, the linear input filter contains unique information about the excitatory and inhibitory population that is not shared with the other 3 attribute sets. Most importantly, the high value of private variance for the linear input filter, along with low residual values compared to other attribute sets indicates that linear input filters are likely to be the most informative attributes for explaining neuronal heterogeneity compared to passive biophysical, action potential, and adaptation current attributes. On the contrary, high values of shared variance for passive biophysical attributes and action potential attributes show that these attributes share a common structure, and can be a good predictor of one another. It is also important to observe that the total variance explained (shared + private) by adaptation currents is lower than action potential attributes for both excitatory and inhibitory populations, suggesting that action potential attributes are more informative about neuronal heterogeneity than adaptation currents.