The Expressive Leaky Memory Neuron: an Efficient and Expressive Phenomenological Neuron Model Can Solve Long-Horizon Tasks

We thank the referee for evaluating our manuscript and for constructive criticism, in particular concerning scientific writing style, earlier introduction of Branch-ELM, and overall limitations of the work. 1) *Using more scientific and neutral wording for clarity:* We agree the manuscript can be improved by replacing and reformulating ambiguous wording (such as the ones mentioned above) to be more scientific and neutral. We updated the manuscript where we found such formulations accordingly. Furthermore, we also introduce the Branch-ELM together with the ELM at the beginning of the manuscript. 2) *Aligning the abstract and contributions formulations for transparency:* Likewise, we concede that some sentences can be improved, we would like to offer the following modification, but are open to further changes to increase alignment: Abstract modifications: - Replace the phrase “the ELM neuron proves to be competitive on both datasets” with “the ELM neuron displays substantial long-range processing capabilities” - Replace the last sentence with “These findings raise further questions about the computational sophistication of individual cortical neurons and their role in extracting long-range temporal dependencies.” Contribution modifications: - 3rd point: The ELM neuron facilitates the formulation and validation of hypotheses regarding the underlying high-level neuronal computations using suitable architectural ablations. - 4th point: Lastly, we demonstrate the considerable long-sequence processing capabilities of the ELM neuron through the use of long memory and synapse timescales. 3) *Other neuron models on NeuronIO and their sizes:* The number of trainable parameters might look deceptively large, but the input dimension is also very large (1278 on the NeuronIO). To make a proper comparison, we followed your suggestion and fitted additional models to the NeuronIO dataset (Table S4, also referenced now in the text). Our simplest model, a Leaky Integrate and Fire (LIF) neuron, has 1280 parameters and achieves 0.9117 AUC, and an Adaptive-LIF (ALIF) neuron, has 1281 and achieves 0.9115 AUC. In comparison, we have found that a Branch-ELM with 5329 parameters ($dm=15$, $d_{tree}=45$ and $d_{brch}=65$) can achieve 0.9915 AUC. Thus, a Branch-ELM neuron with about 4 parameters per input dimension achieves a good fit to the data. 4) *Fitting other neuron models to expand neuroscience contribution:* We also additionally provide results on fits of ELM neurons to LIF and ALIF data (Figure S6). The ELM neuron provides accurate fits with linear integration and few memory units; as expected given the ground truth architectures of those models. 5) *Clarifying the limitations of the model, training and dataset:* We highlight the limitations of the dataset in the main part of the manuscript; namely that we don’t train the ELM neuron in a biologically plausible way (e.g. similar to your suggestion), and that only the SHD and SHD-Adding are bio-inspired tasks, and the NeuronIO task is not. We highlight that the ELM neuron cannot give mechanistic insight into the functioning of synaptic plasticity of cortical neurons, and that other training setups are required to investigate the analogies in more detail. 6) *Future possible research directions:* We agree that exploring whether the ELM neuron would display similar I/O in the case of fitting it using bio-inspired techniques on bio-inspired tasks is an exciting avenue. However, we also agree that it is out of scope for the current manuscript. We plan to explore it in future research.

We thank the referee for evaluating our manuscript and for constructive criticism, in particular with respect to the need to clarify the Branch-ELM design and the used training parameters. 1) *A paragraph defining Branch-ELM would be necessary. Can you elaborate more on the intuition behind this variant? How does it work? When is it more suitable compared to the vanilla ELM? Why is it important to over-sample the input in this case?* We now have a dedicated description of the Branch-ELM at the beginning of the paper, together with the definition of ELM (in “The integration mechanism dynamics” paragraph), and generally expand on its discussion. The intuition: the Branch-ELM is inspired by the biological neuron, whose dendritic tree is composed of smaller hierarchically organized units called dendritic branches. While the dendritic tree overall might compute a highly nonlinear transformation, most inputs are summed linearly, and organization into branches guides where linear and where nonlinear summation is appropriate. We mimic this idea by slicing the input into a number (d_tree) of equally sized “branches” (for our best model, each synapse takes part in four slices, accounting for the mismatch between the ground truth and the branch assignment). Inputs are summed linearly within the slice. However, we let the downstream MLP take care of any further processing nonlinearities. If the slice size is large and the number of branches few, this approach reduces dramatically the number of parameters by reducing the size of MLP. - However, using this stark simplification, we lose general expressivity, and performance is expected to degrade unless the inputs are assigned to the synapse belonging to the correct ground truth branch identity and branch computations can be well approximated by a rectified sum. - As we only know that neighboring inputs mostly belong to the same branch identity, we sampled larger overlapping windows than the actual suspected branch sizes and let the model discard the incorrect assignments by modifying the synaptic weights. - In the appendix Figure S5 we show that randomly assigning synapses to branches or fixing the synaptic weights severely hurts performance, and having the incorrect branch activation like “sigmoid” or “relu” also impairs performance significantly. - In general, we expect the Branch-ELM to work great on datasets with large input dimensionality with low information density, such as SHD or SHD-Adding (coding for one or two digits per sample). We randomly oversample the input for the Branch-ELM on that dataset. 2) *The term "fixed trainable" time constant is confusing. Is it a single tau value learned for each neuron that does not change after training?* We agree that our phrasing was unfortunate, we meant “constant”, because they do not change during the execution, but they can be trained. We write now “constant explicit (trainable) timescales”. 3) *In Appendix B, the general training setup is detailed (batch size of 8, etc.). However, later on, different hyperparameters are used for the datasets. It is not clear where this general training setup was used.* Thank you for catching this inconsistency. There are parameters that can change across tasks (including the optimizer, batch size and initial learning rate). We report them now individually in the corresponding tasks. 4) *Does Figure S6a (now S8) a) show any specific patterns?* Overall the NeuronIO dataset contains mostly random biologically inspired firing patterns as input; this means varying ratios of inhibitory (blue) and excitatory (red) input, varying amount of input overall, varying degrees of input clustering, etc. The particular input pattern nicely shows this variation of excitation and inhibition over the course of two seconds. There is no structure beyond this in the input, unlike e.g. for the SHD or SHD-Adding dataset.

We thank the referee for evaluating our manuscript and for constructive criticism, in particular with respect to expanding on the scalability of the ELM neuron. 1) *Clarification of MLP design choice of ELM neuron:* We will include in the manuscript that one of the key modeling goals of the ELM design was to quantify the degree of nonlinearity necessary to model a neuron’s dynamics; hence the choice of MLP as it offers a straightforward way to do this using $d_{mlp}$ and $l_{mlp}$. In contrast, in more biologically realistic detailed biophysical models, there is no straightforward way to quantify the amount of nonlinearity in such models, and therefore offers less interpretability along this dimension compared to the ELM neuron. 2) *Clarification of saturating performance on NeuronIO:* We will add to the NeuronIO task description that the underlying Biophysical model displays chaotic dynamics, meaning any surrogate model's performance will satire below perfect prediction (see also LSTM and TCN), due to the inherent unpredictability of the task. We additionally provide LIF and ALIF models *(see additional pdf)* as an example of fitting neuron models that do not display chaotic dynamics; the ELM neuron performance saturates close to perfect predictability in this case. 3) *Clarification of tuning ELM parameters:* We will highlight that the ELM neuron primarily only requires tuning along the memory $d_m$ dimension (and $\tau_s$ for very sparse tasks). We add the reference to the text for the hyper-parameters tuning table in the appendix that highlights how other hyperparameters can be derived from $d_m$, the input dimensionality $d_s$, and the task length $T$, and what default parameters should be used otherwise. A further testament to this is the LRA task, where mostly the same hyper-parameters across all tasks. Other training parameters such as batch size, learning rate, etc. still need tuning of course. Therefore, we believe the ELM neuron does not require more tuning than e.g. the LSTM. 4) *Number of Long-range ML tasks:* There might be a misunderstanding concerning the number of tasks. The LRA benchmark contains 6 rather different tasks. Additionally, the Spoken Heidelberg Digits Task (SHD), and particularly SHD-addition, has a context length of 2000 for a bin size of 1, and therefore is twice as long as, e.g., the Image or Pathfinder task from LRA. Therefore, we provide results on a total of 8 long-range tasks. 5) *Addressing when the ELM neuron can be expected to be efficient:* We agree that this needs further clarification. The ELM and in particular Branch-ELM neuron can mainly be expected to be much more efficient if the input dimensionality is very large, such as in the NeuronIO task ($d_s=1278$), or somewhat in the SHD and SHD-Adding task ($d_s=700$), and less so e.g. on LRA ($d_s = 8$ to $d_s = 169)$; also note the increased relative performance for language tasks with higher input dimensionality. We now elaborate this condition for efficiency more clearly in the manuscript. 6) *Addressing the scalability and limitations of the ELM neuron:* We acknowledge the scaling limitation of individual ELM neurons, which we set out to explore using the LRA benchmark, however, we would like to point out that the training was kept fairly simple, and we did not use e.g. warmup, weight decay, layer-norm, dropout, etc (which the comparison models did). Additionally, we suspect that assembling such complex neurons in more sophisticated multi-layered networks might provide the desired scalability; unfortunately, this is beyond the scope of this work. Nevertheless, we provide an ablation of the ELM neuron and S5 model on Pathfinder in terms of parameters in Tables S5 and S6 showing that an ELM with 3.5K parameters is outperforming a single layered S5 with ~170K parameters (62.95% ACC vs 58.19% ACC). Furthermore, motivated by the observation that SOTA models on LRA use various regularization techniques, we train ELM neuron with various dropout probabilities on Pathfinder in Table S7, and find that models trained with 0.3 dropout probability for 1000 Epochs can reach ~82\% accuracy, and improvement of 10% accuracy, evidencing that the performance gap might be reduced through additional proper regularization techniques. 7. *Training with biological learning rules:* We agree that it would be interesting to study the biologically plausible training, and it might happen that the training will converge to a different solution. However, for this manuscript, we aimed to establish how the model can perform with standard machine learning optimizers (that typically define the upper limit on the performance, seldomly achieved by the biologically plausible rules). Finding how to train this model with, for example, Eq.Prop or even more complicated bio-plausible plasticity rules would be an exciting direction. We added it to future work suggestions.

We thank the referee for evaluating our manuscript and for constructive criticism, in particular concerning how the manuscript's presentation can be improved for clarity. 1) *Better introduction and explanation of the Branch-ELM neuron model:* We improve the presentation of our paper by introducing the Branch-ELM together with the ELM neuron in section 2, by additionally expanding on the parameters involved in scaling the MLP output, adding references to the supplementary tables and by clarifying the Branch-ELM processing behavior in detail. 2) *Clarification of the parameters scaling of the MLP output:* The constant $\lambda$ (float constant) and $1-\kappa_m$ (float vector, dependent on $tau_m$) are factors that scale the output of the MLP. Together, they ensure that the memory values $m_t$ are constrained to stay within a range of [-$\lambda$, $\lambda$] if $tanh$ range is [-1,1], which prevents exploding gradients. We now added a short explanation to the formulation of the model (in “The integration mechanism dynamics” section) 3) *Clarification of the Branch-ELM processing of its inputs on NeuronIO:* Indeed, this is correct; if the ordering of input channels is shuffled, performance degrades significantly for the Branch-ELM. Neighboring input channels are typically located next to each other in the biophysical model dataset we are using, hence we oversample the input in a moving window fashion to exploit this insight (the resulting overlap between branches is to account for uncertainty in exact ground truth branch bounds). We tested the shuffling effect in Supplementary Figure S5 (purple trace), where random synapse-to-branch assignment makes the model incapable of reaching the spike-prediction threshold. For other datasets, such as SHD, however, we randomly assign synapses to branches. We will highlight this in the main text and refer to additional branch-related ablations in the appendix. 4) *Clarification of ELM neuron and LSTM performance for larger bin sizes on SHD:* The ELM and Branch-ELM neurons consistently outperform the LSTM for all binning sizes. Nevertheless, performance degrades for larger bin sizes for all models, as the input binning is a lossy pre-processing step, where temporal information is eliminated. Performance improves with the bin size for LSTM because, for the small time bins, LSTM could not take advantage of the whole temporal structure but gets into training problems due to the many time bins it has to keep in memory. The information is lost as the bin size increases, but the small number of bins enables LSTM to learn the task. We show that the temporal structure of the task is not a problem for ELM neurons, so they are only facing the drawbacks of the coarser data discretization. Overall, the observed performance dynamics is not a shortcoming of, e.g. the ELM or LSTM models but a consequence of the data preprocessing. We now discuss it in the main text, and include a statement about the hyper-tuning on the dataset in the appendix. 5) *Correspondence between stacked ELM neuron and biological neuron unclear:* We agree that it is a promising direction of future investigation, particularly with the focus on increasing performance. Our main focus for the current contribution is to explore the performance and limitations of the single expressive neuron. To maintain biological correspondence, it would be interesting to train the recurrent network of ELM neurons (mimicking the recurrent nature of computations in the brain). 6) *Comparison to matched parameter performance on LRA between:* To provide some evaluation results with matched parameters on LRA, we provide an additional ablation on Pathfinder of the number of memory units for the ELM neuron and number of layers for the S5 model respectively; now displayed in Tables S5 and S6 The results show that an ELM with 3.5K parameters is outperforming a single layered S5 with ~170K parameters (62.95% ACC vs 58.19% ACC). 7) “Would this neuron be able to model synapse dynamics such as short-term plasticity (Tsodyks et al. 1998)?” At the moment, the ELM neuron does not have a dedicated mechanism for short-term plasticity. One reason for this is that STD is primarily implemented on the pre-synaptic side (using depletion of synaptic resources). However, we suspect that, in principle, the ELM neuron can capture various short-term plasticity effects dependent solely on internal processes by leveraging a sufficient number of memory units (much larger than used now). We tested the performance of ELM on the simplified neuron with spike-frequency adaptation (aLIF) that is somewhat similar to the STD effect but is completely controlled by the neuron (see Appendix Table S4).