Original: Biologically Inspired Dynamic Thresholds for Spiking Neural Networks
Revised: Biologically Inspired Dynamic Thresholds for Spiking Neural Networks

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: The dynamic membrane potential threshold, as one of the essential properties of abiological neuron, is a spontaneous regulation mechanism that maintains neuronalhomeostasis, i .
Revised: The dynamic membrane potential threshold, as one of the essential properties of a biological neuron, is a spontaneous regulation mechanism that maintains neuronal homeostasis, i .

Original: e ., the constant overall spiking firing rate of a neuron.
Revised: e ., the constant overall spiking firing rate of a neuron.

Original: As such, theneuron firing rate is regulated by a dynamic spiking threshold, which has beenextensively studied in biology.
Revised: As such, the neuron firing rate is regulated by a dynamic spiking threshold, which has been extensively studied in biology.

Original: Existing work in the machine learning community does not employ bioplausible spiking threshold schemes.
Revised: Existing work in the machine learning community does not employ bioinspired spiking threshold schemes.

Original: This work aimsat bridging this gap by introducing a novel bioinspired dynamic energy-temporalthreshold (BDETT) scheme for spiking neural networks (SNNs).
Revised: This work aims at bridging this gap by introducing a novel bioinspired dynamic energy-temporal threshold (BDETT) scheme for spiking neural networks (SNNs).

Original: The proposed
Revised: The proposed

Original: BDETT scheme mirrors two bioplausible observations: a dynamic threshold has1) a positive correlation with the average membrane potential and 2) a negativecorrelation with the preceding rate of depolarization.
Revised: BDETT scheme mirrors two bioplausible observations: a dynamic threshold has 1) a positive correlation with the average membrane potential and 2) a negative correlation with the preceding rate of depolarization.

Original: We validate the effectivenessof the proposed BDETT on robot obstacle avoidance and continuous control tasksunder both normal conditions and various degraded conditions, including noisyobservations, weights, and dynamic environments.
Revised: We validate the effectiveness of the proposed BDETT on robot obstacle avoidance and continuous control tasks under both normal conditions and various degraded conditions, including noisy observations, weights, and dynamic environments.

Original: We find that the BDETT outperforms existing static and heuristic threshold approaches by significant marginsin all tested conditions, and we confirm that the proposed bioinspired dynamicthreshold scheme offersbioplausible homeostasis to SNNs in complex real-worldtasks.
Revised: We find that the BDETT outperforms existing static and heuristic threshold approaches by significant margins in all tested conditions, and we confirm that the proposed bioinspired dynamic threshold scheme offers homeostasis to SNNs in complex real-world tasks.

Original:  Introduction
Revised:  Introduction

Original: A spiking neural network (SNN) is a bioinspired neural network.
Revised: A spiking neural network (SNN) is a bioinspired neural network.

Original: Each spiking neuron is a mathematical model abstracted from the properties of a biological neuron.
Revised: Each spiking neuron is a mathematical model abstracted from the properties of a biological neuron.

Original: Spiking neurons communicatewith each other through spike trains, mimicking the information transfer process of biological neurons [1, 2, 3].
Revised: Spiking neurons communicate with each other through spike trains, mimicking the information transfer process of biological neurons [1, 2, 3].

Original: Similar to how biological action potentials are all-or-none impulses, the spikes of SNNsare commonly binary voltage pulses.
Revised: Similar to how biological action potentials are all-or-none impulses, the spikes of SNNs are commonly binary voltage pulses.

Original: Leveraging this binary representation, specifically designedneuromorphic hardware [4, 5, 6], e . g .,
Revised: Leveraging this binary representation, specifically designed neuromorphic hardware [4, 5, 6], e . g .,

Original: TrueNorth [7] and Loihi [8], can run SNNs at extremely lowpower levels; they are 75 times more energy-efficient than their deep neural network counterparts onlow-power GPU platforms [9].
Revised: TrueNorth [7] and Loihi [8], can run SNNs at extremely low power levels; they are 75 times more energy-efficient than their deep neural network counterparts on low-power GPU platforms [9].

Original: As such, recently, SNNs have rapidly emerged as effective models forrobotic control tasks, especially in mobile robots that demand low power consumption [10, 11].
Revised: As such, recently, SNNs have rapidly emerged as effective models for robotic control tasks, especially in mobile robots that demand low power consumption [10, 11].

Original: However, existing SNNs suffer from poor generalizability, unlike their biological counterparts.
Revised: However, existing SNNs suffer from poor generalizability, unlike their biological counterparts.

Original: Biologically, a neuron leverages a spontaneous regulation mechanism to maintain neuronal homeostasis [12]—the stable overall spiking firing rate or excitability within a network [13]—to robustlyadapt to different external conditions and offer strong generalization.
Revised: Biologically, a neuron leverages a spontaneous regulation mechanism to maintain neuronal homeostasis [12]—the stable overall spiking firing rate or excitability within a network [13]—to robustly adapt to different external conditions and offer strong generalization.

Original: A dynamic threshold, onetype of regulatory mechanism, plays an essential role in maintaining neuronal homeostasis by regulating the action potential firing rate; such thresholds are widely observed in different nervoussystems [14, 15, 16, 17, 18, 19, 20, 21, 22, 23].
Revised: A dynamic threshold, one type of regulatory mechanism, plays an essential role in maintaining neuronal homeostasis by regulating the action potential firing rate; such thresholds are widely observed in different nervous systems [14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

Original: This threshold can be regarded as an adaptation tomembrane potentials at short timescales [16], and it influences how the received signals of a neuronare encoded into a spike.
Revised: This threshold can be regarded as an adaptation to membrane potentials at short timescales [16], and it influences how the received signals of a neuron are encoded into a spike.

Original: Even though different dynamic threshold schemes have been observed and extensively studied inneuroscience, only a handful of existing works investigate bioinspired dynamic threshold rules toimprove the generalization of SNNs.
Revised: Even though different dynamic threshold schemes have been observed and extensively studied in neuroscience, only a handful of existing works investigate bioinspired dynamic threshold rules to improve the generalization of SNNs.

Original: Hao et al .
Revised: Hao et al .

Original: [24] proposed a dynamic threshold method thatrelies on a heuristic dynamic scaling factor to gradually slow the growth of a threshold.
Revised: [24] proposed a dynamic threshold method that relies on a heuristic dynamic scaling factor to gradually slow the growth of a threshold.

Original: Conversely,instead of controlling threshold growth, Shaban et al .
Revised: Conversely, instead of controlling threshold growth, Shaban et al .

Original: [25] leveraged double exponential functionsto manage the threshold decay.
Revised: [25] leveraged double exponential functions to manage the threshold decay.

Original: Kim et al .
Revised: Kim et al .

Original: [26] used a predefined target firing count to adjust theirthreshold but did not define the optimal target firing count.
Revised: [26] used a predefined target firing count to adjust their threshold but did not define the optimal target firing count.

Original: No existing work has demonstrated that abioinspired dynamic threshold scheme can achieve homeostasis in real-world tasks.
Revised: No existing work has demonstrated that a bioinspired dynamic threshold scheme can achieve homeostasis in real-world tasks.

Original: More importantly,the existing work only validates the proposed dynamic threshold rules under ideal normal conditionswithout testing generalization to degraded conditions, which we argue is essential to validate whetherhomeostasis is achieved or not.
Revised: More importantly, the existing work only validates the proposed dynamic threshold rules under ideal normal conditions without testing generalization to degraded conditions, which we argue is essential to validate whether homeostasis is achieved or not.

Original: The direct use of bioplausible models in SNNs remains challenging, as most of these models are basedon single cells in the nervous system and contain many optimized constants.
Revised: The direct use of bioplausible models in SNNs remains challenging, as most of these models are based on single cells in the nervous system and contain many optimized constants.

Original: In this work, we lift thislimitation and introduce a novelbioinspired dynamic energy-temporal threshold (BDETT) scheme for
Revised: In this work, we lift this limitation and introduce a novel dynamic energy-temporal threshold (BDETT) scheme for

Original: SNNs; the scheme comprises two components: a dynamic energy threshold and a dynamic temporalthreshold schema.
Revised: SNNs; the scheme comprises two components: a dynamic energy threshold and a dynamic temporal threshold schema.

Original: The two components reflect the following two biological observations: in vivo , thedynamic threshold exhibits a positive correlation with the average membrane potential and a negativecorrelation with the preceding rate of depolarization ( i .
Revised: The two components reflect the following two biological observations: in vivo , the dynamic threshold exhibits a positive correlation with the average membrane potential and a negative correlation with the preceding rate of depolarization ( i .

Original: e ., the excitatory status)
Revised: e ., the excitatory status)

Original: The dynamicenergy threshold is inspired by a biological predictive model which can predict the occurrences ofspikes based on the previous membrane potential in the inferior colliculus of a barn owl [16].
Revised: The dynamic energy threshold is inspired by a biological predictive model which can predict the occurrences of spikes based on the previous membrane potential in the inferior colliculus of a barn owl [16].

Original: Theproposed dynamic temporal threshold component is inspired by the fact that a monoexponentialfunction can effectively present a negative correlation [17, 22].
Revised: The proposed dynamic temporal threshold component is inspired by the fact that a monoexponential function can effectively present a negative correlation [17, 22].

Original: Notably, we provide an analysis of theoriginal biological models and propose layerwise statistical cues for SNNs to replace the constants inthe two original biological models.
Revised: Notably, we provide an analysis of the original biological models and propose layerwise statistical cues for SNNs to replace the constants in the two original biological models.

Original: We integrate the proposed BDETT into two widely used SNN models: a spike response model(SRM) [27] and a leaky integrate-and-fire (LIF) model [28] (see Supplementary Material for details).
Revised: We integrate the proposed BDETT into two widely used SNN models: a spike response model (SRM) [27] and a leaky integrate-and-fire (LIF) model [28].

Original: The effectiveness of BDETT is validated with these two SNN models forboth autonomous roboticobstacle avoidance and continuous control  tasks under normal and various degraded conditions, e .
Revised: The effectiveness of BDETT is validated with these two SNN models for autonomous robotic obstacle avoidance, continuous control and image classification tasks under normal and various degraded conditions, e .

Original: g .,dynamic obstacles, noisy inputs, and weight uncertainty.
Revised: g ., dynamic obstacles, noisy inputs, and weight uncertainty.

Original: Extensive experimental results validate thatthe SNNs equipped with the proposed BDETT offer the strongest generalization across all testedscenarios.
Revised: Extensive experimental results validate that the SNNs equipped with the proposed BDETT offer the strongest generalization across all tested scenarios.

Original: More importantly, we quantitatively validate that BDETT can significantly increase thehomeostasis of the host SNN.
Revised: More importantly, we quantitatively validate that BDETT can significantly increase the homeostasis of the host SNN for robotic control tasks.

Original: This is the first work to demonstrate thatbioinspired dynamic thresholdschemes can offer bioplausible homeostasis to SNNs in robotic real-world tasks under normal anddegraded conditions, dramatically enhancing the generalizability and adaptability of the host SNNs.
Revised: This is the first work to demonstrate that dynamic threshold schemes can offer bioplausible homeostasis to SNNs in robotic real-world tasks under normal and degraded conditions, dramatically enhancing the generalizability and adaptability of the host SNNs.

Original: In particular, we make the following contributions in this work:
Revised: In particular, we make the following contributions in this work:

Original: • We introduce a bioinspired dynamic threshold scheme for SNNs that increases their generalizability.• We devise a method that uses layerwise statistical cues of SNNs to set the parameters of ourbioinspired threshold method.•
Revised: • We introduce a bioinspired dynamic threshold scheme for SNNs that increases their generalizability. • We devise a method that uses layerwise statistical cues of SNNs to set the parameters of our bioinspired threshold method.

Original:  We validate that the proposed threshold scheme achieves bioplausible homeostasis, dramatically enhancing the generalizability across tasks, including obstacle avoidance and roboticcontrol, and in normal and degraded conditions.
Revised: • We validate that the proposed threshold scheme achieves bioplausible homeostasis, dramatically enhancing the generalizability across tasks, including obstacle avoidance and robotic control, and in normal and degraded conditions.

Original: Scope We propose a novel approach to setting the parameters of our threshold scheme using layerwisestatistical cues of an SNN.
Revised: Scope We propose a novel approach to setting the parameters of our threshold scheme using layerwise statistical cues of an SNN.

Original: Although this is essential for the proposed method to be effective,implementing these statitical blocks directly in neuromorphic hardware may require extra engineeringefforts, which is out of the scope of this work.
Revised: Although this is essential for the proposed method to be effective, implementing these statitical blocks directly in neuromorphic hardware may require extra engineering efforts, which is out of the scope of this work.

Original:  Background and Related Work2.1 Spiking Neural Networks (SNNs)
Revised:  Background and Related Work 2.1 Spiking Neural Networks (SNNs)

Original: Various models for spiking neurons have been described to mathematically describe the properties ofa nervous neuron.
Revised: Various models for spiking neurons have been described to mathematically describe the properties of a nervous neuron.

Original: Typically, three conditions are considered by these models: resting, depolarization,and hyperpolarization.
Revised: Typically, three conditions are considered by these models: resting, depolarization, and hyperpolarization.

Original: When a neuron is resting, it maintains a constant membrane potential.
Revised: When a neuron is resting, it maintains a constant membrane potential.

Original: Thechange in membrane potential can be either a decrease or an increase relative to the resting potential.
Revised: The change in membrane potential can be either a decrease or an increase relative to the resting potential.

Original: An increase in the membrane potential is called depolarization, which enhances the ability of a cell to generate an action potential; it is excitatory.
Revised: An increase in the membrane potential is called depolarization, which enhances the ability of a cell to generate an action potential; it is excitatory.

Original: In contrast, hyperpolarization describes a reduction inthe membrane potential, which makes the associated cell less likely to generate an action potential,and, as such, is inhibitory.
Revised: In contrast, hyperpolarization describes a reduction in the membrane potential, which makes the associated cell less likely to generate an action potential, and, as such, is inhibitory.

Original: All inputs and outputs of a spiking neuron model are sequences of spikes.
Revised: All inputs and outputs of a spiking neuron model are sequences of spikes.

Original: A sequence of spikes is called a spike train and is defined as s ( t ) =
Revised: A sequence of spikes is called a spike train and is defined as s ( t ) =

Original: Σ t ( f ) ∈F δ ( t − t ( f ) ) , where Frepresents the set of times at which the individual spikes occur [29].
Revised: Σ t ( f ) ∈F δ ( t − t ( f ) ) , where F represents the set of times at which the individual spikes occur [29].

Original: Typical spiking neuron modelsset the resting potential as 0 .
Revised: Typical spiking neuron models set the resting potential as 0 .

Original: However, existing models achieve depolarization and hyperpolarizationin substantially different ways.
Revised: However, existing models achieve depolarization and hyperpolarization in substantially different ways.

Original: In the following, we briefly review two commonly used models: thespike response model (SRM) [27] and leaky integrate-and-fire (LIF) model [28].
Revised: In the following, we briefly review two commonly used models: the spike response model (SRM) [27] and leaky integrate-and-fire (LIF) model [28].

Original: More details aboutthese two models are provided in Supplementary Note 1.
Revised: More details about these two models are provided in Supplementary Note 1.

Original: Spike Response Model (SRM)
Revised: Spike Response Model (SRM)

Original: An SRM first converts an incoming spike train s i ( t ) into a spikeresponse signal as ( ε ∗ s i )( t ) , where ε ( · ) is a spike response kernel.
Revised: An SRM first converts an incoming spike train s i ( t ) into a spike response signal as ( ε ∗ s i )( t ) , where ε ( · ) is a spike response kernel.

Original: Then, the generated spikeresponse signal is scaled by a synaptic weight w i .
Revised: Then, the generated spike response signal is scaled by a synaptic weight w i .

Original: Depolarization is achieved by summing all thescaled spike response signals: Σ i w i ( ε ∗ s i )( t ) .
Revised: Depolarization is achieved by summing all the scaled spike response signals: Σ i w i ( ε ∗ s i )( t ) .

Original: When incoming spike trains trigger a spike s ( t ) ,the SRM models hyperpolarization by defining a refractory potential as ( ζ ∗ s )( t ) , where ζ ( · ) is arefractory kernel.
Revised: When incoming spike trains trigger a spike s ( t ) , the SRM models hyperpolarization by defining a refractory potential as ( ζ ∗ s )( t ) , where ζ ( · ) is a refractory kernel.

Original: Leaky Integrate-and-Fire (LIF)
Revised: Leaky Integrate-and-Fire (LIF)

Original: An LIF model is a simplified variant of an SRM.
Revised: An LIF model is a simplified variant of an SRM.

Original: This schemedirectly processes incoming spike trains and ignores the spike response kernel.
Revised: This scheme directly processes incoming spike trains and ignores the spike response kernel.

Original: Hyperpolarization isachieved by a simplified step decay function, f d ( s ( t ))
Revised: Hyperpolarization is achieved by a simplified step decay function, f d ( s ( t ))

Original: =
Revised: =

Original: D for s ( t ) = 0; 0 for s ( t ) = 1 .
Revised: D for s ( t ) = 0; 0 for s ( t ) = 1 .

Original: 2.2 Spiking Neural Networks for Robot Control
Revised: 2.2 Spiking Neural Networks for Robot Control

Original: Biological neural circuits have an impressive ability to avoid obstacles robustly in complex dynamical environments, e . g ., as in dragonfly flight trajectories.
Revised: Biological neural circuits have an impressive ability to avoid obstacles robustly in complex dynamical environments, e . g ., as in dragonfly flight trajectories.

Original: Inspired by this observation, recently,researchers have explored SNNs for obstacle avoidance [30, 31, 32, 33].
Revised: Inspired by this observation, recently, researchers have explored SNNs for obstacle avoidance [30, 31, 32, 33].

Original: For example, Tang et al .
Revised: For example, Tang et al .

Original: [33]devised an SNN to mimic a neurophysiologically plausible connectome of the brain’s navigationalsystem without assuming all-to-all connectivity.
Revised: [33] devised an SNN to mimic a neurophysiologically plausible connectome of the brain’s navigational system without assuming all-to-all connectivity.

Original: Following the path, Tang et al .
Revised: Following the path, Tang et al .

Original: [9] proposed aspiking deep deterministic policy gradient (SDDPG) method to train a LIF-based spiking actornetwork (SAN) for mapless navigation.
Revised: [9] proposed a spiking deep deterministic policy gradient (SDDPG) method to train a LIF-based spiking actornetwork (SAN) for mapless navigation.

Original: They show that SNNs can robustly control a robot in mappingtasks while being able to explore an unknown environment.
Revised: They show that SNNs can robustly control a robot in mapping tasks while being able to explore an unknown environment.

Original: SNNs have also been proposed forcontinuous robot control tasks.
Revised: SNNs have also been proposed for continuous robot control tasks.

Original: Patel et al .
Revised: Patel et al .

Original: [34] proposed to combine SNNs with a Deep Q-networkalgorithm, improving the robustness to occlusion in the input image.
Revised: [34] proposed to combine SNNs with a Deep Q-network algorithm, improving the robustness to occlusion in the input image.

Original: Tang et al .
Revised: Tang et al .

Original: [35] proposed apopulation-coded spiking actor network (PopSAN) to solve high-dimensional continuous controlproblems, trained using deep reinforcement learning algorithms.
Revised: [35] proposed a population-coded spiking actor network (PopSAN) to solve high-dimensional continuous control problems, trained using deep reinforcement learning algorithms.

Original: Recently, modern neuromorphichardware has made it possible to deploy SNNs on neuromorphic processors in ultra-low powerenvelopes [9, 36, 37, 38].
Revised: Recently, modern neuromorphic hardware has made it possible to deploy SNNs on neuromorphic processors in ultra-low power envelopes [9, 36, 37, 38].

Original: Compared to existing convolutional deep policy networks [39] on themobile-GPUs such as the Nvidia Jetson TX2, SAN and PopSAN on Loihi neuromorphic processorconsume 75 and 140 times less energy per inferences, respectively.
Revised: Compared to existing convolutional deep policy networks [39] on the mobile-GPUs such as the Nvidia Jetson TX2, SAN and PopSAN on Loihi neuromorphic processor consume 75 and 140 times less energy per inferences, respectively.

Original: All SNN-based models discussedabove only consider static spiking thresholds.
Revised: All SNN-based models discussed above only consider static spiking thresholds.

Original: More importantly, experiments show that they sufferfrom poor generalization and fail in realistic degraded conditions.
Revised: More importantly, experiments show that they suffer from poor generalization and fail in realistic degraded conditions.

Original: In this work, we use both SAN and
Revised: In this work, we use both SAN and

Original: PopSAN as testbeds and baseline methods to validate the effectiveness of the proposed bioinspireddynamic threshold scheme, BDETT.
Revised: PopSAN as testbeds and baseline methods to validate the effectiveness of the proposed bioinspired dynamic threshold scheme, BDETT.

Original:  Bio-Plausible Dynamic Energy-Temporal Threshold (BDETT)
Revised:  Bioinspired Dynamic Energy-Temporal Threshold (BDETT)

Original: Motivated by the behavior of spiking threshold dynamics in biological nervous systems, we proposea model with dynamic thresholds that exhibit positive and negative correlations with the averagemembrane potential and the preceding rate of depolarization, respectively.
Revised: Motivated by the behavior of spiking threshold dynamics in biological nervous systems, we propose a model with dynamic thresholds that exhibit positive and negative correlations with the average membrane potential and the preceding rate of depolarization, respectively.

Original: To achieve this behaviorin the proposed scheme, given the i -th neuron in the l -th layer at timestamp t + 1 , we define abioplausible dynamic threshold Θ li ( t + 1) as
Revised: To achieve this behavior in the proposed scheme, given the i -th neuron in the l -th layer at timestamp t + 1 , we define a dynamic threshold Θ li ( t + 1) as

Original: [Equation]
Revised: [Equation]

Original:  where E li ( t ) is the dynamic energy threshold (DET) of the neuron for ensuring a positive correlation,and T li ( t + 1) is the dynamic temporal threshold (DTT), which ensures a negative correlation; see
Revised: (1) where E li ( t ) is the dynamic energy threshold (DET) of the neuron for ensuring a positive correlation, and T li ( t + 1) is the dynamic temporal threshold (DTT), which ensures a negative correlation; see

Original: Figure 1a.
Revised: Figure 1a.

Original: Note that each neuron has a different dynamic threshold at timestamp t + 1 based on theproposed DET and DTT, which we describe below.
Revised: Note that each neuron has a different dynamic threshold at timestamp t + 1 based on the proposed DET and DTT, which we describe below.

Original: Dynamic Energy Threshold (DET) Positive correlations between dynamic thresholds and averagemembrane potentials have been observed in several areas of diverse biological nervous systems, suchas the visual cortex and auditory midbrain [17, 21, 22].
Revised: Dynamic Energy Threshold (DET) Positive correlations between dynamic thresholds and average membrane potentials have been observed in several areas of diverse biological nervous systems, such as the visual cortex and auditory midbrain [17, 21, 22].

Original: With sufficient voltage measurements at spike onsets, one can fit a model to directly predict the voltage of a threshold [40].
Revised: With sufficient voltage measurements at spike onsets, one can fit a model to directly predict the voltage of a threshold [40].

Original: However, thefitted biological model is only meaningful to a specific nervous system, and stimulus or measurementuncertainty can significantly impact the model accuracy.
Revised: However, the fitted biological model is only meaningful to a specific nervous system, and stimulus or measurement uncertainty can significantly impact the model accuracy.

Original: Fontaine et al .
Revised: Fontaine et al .

Original: [16] proposed a biologicalpredictive approach to assess the occurrence of spikes based on the previous membrane potential; thismethod does not rely on voltage measurements at spike onsets.
Revised: [16] proposed a biological predictive approach to assess the occurrence of spikes based on the previous membrane potential; this method does not rely on voltage measurements at spike onsets.

Original: Even though the model was based ona barn owl’s inferior colliculus, it exhibits great generality in terms of threshold variability statisticswith other nervous systems ( e .
Revised: Even though the model was based on a barn owl’s inferior colliculus, it exhibits great generality in terms of threshold variability statistics with other nervous systems ( e .

Original: g ., cortical neurons)
Revised: g ., cortical neurons)

Original: The proposed dynamic energy thresholdis inspired by this biological predictive model but includes several changes that are critical for themodel to be effective in SNNs.
Revised: The proposed dynamic energy threshold is inspired by this biological predictive model but includes several changes that are critical for the model to be effective in SNNs.

Original: For the i -th neuron in the l -th layer at timestamp t , we define
Revised: For the i -th neuron in the l -th layer at timestamp t , we define

Original: [Equation]
Revised: [Equation]

Original: [Equation] where v li ( t ) is the neuron postsynaptic membrane potential at timestamp t ; µ is the mean operator; n lis the total number of neurons in the l -th layer; and η and ψ are two hyperparameters, which are setempirically.
Revised: where v li ( t ) is the neuron postsynaptic membrane potential at timestamp t ; µ is the mean operator; n l is the total number of neurons in the l -th layer; and η and ψ are two hyperparameters, which are set empirically.

Original: Figure 1b shows two example graphs for Eq. 2; η controls the shallow slope, and 1 ψ + ηdefines the slope of the steep part.
Revised: Figure 1b shows two example graphs for Eq. 2; η controls the shallow slope, and 1 ψ + η defines the slope of the steep part.

Original: Intuitively, V lθ ( t ) and V lm ( t ) define a critical region.
Revised: Intuitively, V lθ ( t ) and V lm ( t ) define a critical region.

Original: When the membrane potential v li ( t ) is smallerthan V lm ( t ) , the function has a shallower slope, and the threshold value is dominated by V lθ ( t ) .
Revised: When the membrane potential v li ( t ) is smaller than V lm ( t ) , the function has a shallower slope, and the threshold value is dominated by V lθ ( t ) .

Original: Inthe opposite case, the energy threshold has a higher rate of increase to inhibit a high spiking firingrate.
Revised: In the opposite case, the energy threshold has a higher rate of increase to inhibit a high spiking firing rate.

Original: In the biological predictive model proposed by Fontaine et al.
Revised: In the biological predictive model proposed by Fontaine et al.

Original: [16], V lm ( t ) and V lθ ( t ) are theconstants to be optimized during the model fitting process.
Revised: [16], V lm ( t ) and V lθ ( t ) are the constants to be optimized during the model fitting process.

Original: However, we find that directly adoptingthese two fitted constants in an SNN does not result in generalization.
Revised: However, we find that directly adopting these two fitted constants in an SNN does not result in generalization; see section 4.4.

Original: Refer to experiment in the mainpaper which shows this To tackle this challenge, we leverage the statistical cues of SNN layers toadjust these two important parameters, as defined in Eqs.
Revised: To tackle this challenge, we leverage the statistical cues of SNN layers to adjust these two important parameters, as defined in Eqs.

Original: 3 and 4.
Revised: 3 and 4.

Original: Specifically, we model V lm ( t ) asthe mean of the membrane potentials of the neurons in the layer l .
Revised: Specifically, we model V lm ( t ) as the mean of the membrane potentials of the neurons in the layer l .

Original: The mean value is shifted by abias, 0 .
Revised: The mean value is shifted by a bias, 0 .

Original: 2(max( v li ( t ))
Revised: 2(max( v li ( t ))

Original: − min( v li ( t ))) , which is based on the range of the potentials; see Eq. 3.
Revised: − min( v li ( t ))) , which is based on the range of the potentials; see Eq. 3.

Original: Themotivation behind this formulation is that we aim to couple the DET and the potentials of all otherneurons in the same layer.
Revised: The motivation behind this formulation is that we aim to couple the DET and the potentials of all other neurons in the same layer.

Original: Furthermore, we leverage the bias term to adjust the DET sensitivity to thelayerwise potential range.
Revised: Furthermore, we leverage the bias term to adjust the DET sensitivity to the layerwise potential range.

Original: Here, V lθ ( t ) is modeled based on similar insights, where we use thresholdpotentials ( i .
Revised: Here, V lθ ( t ) is modeled based on similar insights, where we use threshold potentials ( i .

Original: e ., Θ li ( t ) ) instead of membrane potentials; see Eq. 
Revised: e ., Θ li ( t ) ) instead of membrane potentials; see Eq. 4.

Original: We note that the performance ofthe proposed BDETT is not sensitive to the constant value 0 .
Revised: We note that the performance of the proposed BDETT is not sensitive to the constant value 0 .

Original: 2 ; see Supplementary Note 7 for details.
Revised: 2 ; see Supplementary Note 7 for details.

Original: Dynamic Temporal Threshold (DTT)
Revised: Dynamic Temporal Threshold (DTT)

Original: We propose a DTT scheme to address the observed negativecorrelation between the spiking threshold and the preceding rate of depolarization.
Revised: We propose a DTT scheme to address the observed negative correlation between the spiking threshold and the preceding rate of depolarization.

Original: Azouz et al .[17, 22] discovered that a monoexponential function y =
Revised: Azouz et al . [17, 22] discovered that a monoexponential function y =

Original: a + be − V/C can effectively capture thenegative correlation of a biological neuron, where V = dV m /dt ; C is a decay constant; and a , b , and C are parameters to optimize.
Revised: a + be − V/C can effectively capture the negative correlation of a biological neuron, where V = dV m /dt ; C is a decay constant; and a , b , and C are parameters to optimize.

Original: The authors applied this function to 42 cortical neurons andfound significant correlations in 92% of the trials [17].
Revised: The authors applied this function to 42 cortical neurons and found significant correlations in 92% of the trials [17].

Original: We propose a variant of this mechanism.
Revised: We propose a variant of this mechanism.

Original: Inparticular, we replace the constant a with an exponential decay function, and we base the decay rateon the mean of the dynamic thresholds of all neurons in the l -th layer at the previous timestamp t ; bis set to 1 .
Revised: In particular, we replace the constant a with an exponential decay function, and we base the decay rate on the mean of the dynamic thresholds of all neurons in the l -th layer at the previous timestamp t ; b is set to 1 .

Original: Additionally, we empirically set the delay constant C .
Revised: Additionally, we empirically set the delay constant C .

Original: Mathematically, for the i -th neuronin the l -th layer, the DTT at timestamp t + 1 is defined as
Revised: Mathematically, for the i -th neuron in the l -th layer, the DTT at timestamp t + 1 is defined as

Original: [Equation]
Revised: [Equation]

Original: Figure 1c shows two example graphs for Eq.
Revised: Figure 1c shows two example graphs for Eq.

Original: These plots highlight that higher depolarization( i .
Revised: These plots highlight that higher depolarization ( i .

Original: e ., v li ( t + 1) − v li ( t ) > 0 ) leads to a lower temporal threshold, while higher hyperpolarization ( i .
Revised: e ., v li ( t + 1) − v li ( t ) > 0 ) leads to a lower temporal threshold, while higher hyperpolarization ( i .

Original: e .,v li ( t + 1) − v li ( t ) < 0 ) significantly increases the temporal threshold.
Revised: e ., v li ( t + 1) − v li ( t ) < 0 ) significantly increases the temporal threshold.

Original: We propose modeling a similarto how V lm ( t ) is modeled in the DET, that is, by coupling the DTT value and the layerwise dynamicthresholds at the previous timestamp t ( i .
Revised: We propose modeling a similar to how V lm ( t ) is modeled in the DET, that is, by coupling the DTT value and the layerwise dynamic thresholds at the previous timestamp t ( i .

Original: e ., Θ li ( t ) ).
Revised: e ., Θ li ( t ) ).

Original: The delay constant C adjusts the sensitivity ofthe DTT to changes in the temporal potential of a neuron.
Revised: The delay constant C adjusts the sensitivity of the DTT to changes in the temporal potential of a neuron.

Original: As shown in Figure 1c, a lower C valueresults in a substantially faster drop in the DTT value ( i .
Revised: As shown in Figure 1c, a lower C value results in a substantially faster drop in the DTT value ( i .

Original: e ., the black curve) than that provided by ahigher C value ( i .
Revised: e ., the black curve) than that provided by a higher C value ( i .

Original: e ., the blue curve).
Revised: e ., the blue curve).

Original: Interaction of DET and DTT A critical difference between DET and DTT lies in the drivers ofthe two threshold schemes.
Revised: Interaction of DET and DTT A critical difference between DET and DTT lies in the drivers of the two threshold schemes.

Original: DET leverages the magnitude of the membrane potential to estimatea threshold, while the DTT based on the preceding rate of depolarization.
Revised: DET leverages the magnitude of the membrane potential to estimate a threshold, while the DTT based on the preceding rate of depolarization.

Original: Therefore, they may becounteracting or helping each other to achieve an optimal threshold.
Revised: Therefore, they may be counteracting or helping each other to achieve an optimal threshold.

Original: One example is that when noisecauses low potential fluctuations, the overall threshold should increase to suppress the noise.
Revised: One example is that when noise causes low potential fluctuations, the overall threshold should increase to suppress the noise.

Original: In thiscase, the DET increases as the noise increases the membrane potential.
Revised: In this case, the DET increases as the noise increases the membrane potential.

Original: However, DTT remains at arelatively constant threshold ( i .
Revised: However, DTT remains at a relatively constant threshold ( i .

Original: e ., a + 1 ) as the preceding rate of depolarization caused by the noiseis close to 0 .
Revised: e ., a + 1 ) as the preceding rate of depolarization caused by the noise is close to 0 .

Original: When a neuron experiences a fast membrane potential drop, e .
Revised: When a neuron experiences a fast membrane potential drop, e .

Original: g .
Revised: g .

Original: , during the relativerefractory period, we expect the overall threshold to increase.
Revised: , during the relative refractory period, we expect the overall threshold to increase.

Original: In this scenario, even though DETdecreases with the reduced membrane potential, DTT increases faster.
Revised: In this scenario, even though DET decreases with the reduced membrane potential, DTT increases faster.

Original: Hence, the proposed methodincreases the overall threshold in this case.
Revised: Hence, the proposed method increases the overall threshold in this case.

Revised: Please see Supplementary Note 10 for details.

Original:  Experiments
Revised:  Experiments

Original: We assess the effectiveness of BDETT on two different tasks: robot obstacle avoidance and roboticcontinuous control.
Revised: We assess the effectiveness of BDETT on three different tasks: robot obstacle avoidance, robotic continuous control and image classification.

Original: In the robot obstacle avoidance task, a robot aims to reach a randomly chosendestination without touching any obstacle within 1000 steps, counted as a “pass”.
Revised: In the robot obstacle avoidance task, a robot aims to reach a randomly chosen destination without touching any obstacle within 1000 steps, counted as a “pass”.

Original: For this task,we assess methods by measuring success rate (SR), the percentage of successful passes out of 200trials.
Revised: For this task, we assess methods by measuring success rate (SR), the percentage of successful passes out of 200 trials.

Original: As continuous control tasks, we evaluate the HalfCheetah-v3 and Ant-v3 control outputs (see Figure 3a) from the OpenAI gym [41].
Revised: As continuous control tasks, we evaluate the HalfCheetah-v3 and Ant-v control outputs (see Figure 3a) from the OpenAI gym [41].

Original: In these two continuous control tasks, an agent relies on alearned SNN-based control policy to decide the next action based on the current observation ( i .
Revised: In these two continuous control tasks, an agent relies on a learned SNN-based control policy to decide the next action based on the current observation ( i .

Original: e .,state), and each action is associated with a reward; see Figure 3a.
Revised: e ., state), and each action is associated with a reward; see Figure 3a.

Original: We assess control policies with thetotal sum of the rewards.
Revised: We assess control policies with the total sum of the rewards.

Original: Note that the Ant-v3 control task is more challenging than HalfCheetah-v3,with significantly large state and action spaces.
Revised: Note that the Ant-v3 control task is more challenging than HalfCheetah-v3, with significantly large state and action spaces.

Revised: Top-1 classification accuracy is used to assess image classification.

Original: In addition to evaluating the control output, i .
Revised: For the robotic control tasks, in addition to evaluating the control output, i .

Original: e ., SR and total reward, we also measure the homeostasisof the host SNNs.
Revised: e ., SR and total reward, we also measure the homeostasis of the host SNNs.

Original: In particular, we use three statistical metrics, FR m , FR mstd , and FR sstd , to quantifythe homeostasis of an SNN; these metrics are based on the neuron firing rate.
Revised: In particular, we use three statistical metrics, FR m , FR mstd , and FR sstd , to quantify the homeostasis of an SNN; these metrics are based on the neuron firing rate.

Original: FR m is the meanneuron firing rate of an SNN across all P trials; FR mstd is the average of P standard deviations, andeach of them is the standard deviation of the neuron firing rates of an SNN during a single trial; FR sstddenotes the standard deviation of the P standard deviations.
Revised: FR m is the mean neuron firing rate of an SNN across all P trials; FR mstd is the average of P standard deviations, and each of them is the standard deviation of the neuron firing rates of an SNN during a single trial; FR sstd denotes the standard deviation of the P standard deviations.

Original: FR sstd represents the standard deviationacross all P trials, while FR mstd denotes the mean of these standard deviations.
Revised: FR sstd represents the standard deviation across all P trials, while FR mstd denotes the mean of these standard deviations.

Original: Details on these threemetrics can be found in Supplementary Note 2.
Revised: Details on these three metrics can be found in Supplementary Note 2.

Original: Experimental Setup For robot obstacle avoidance tasks, the we use variants of the spiking actornetwork (SAN) [9] as host SNN.
Revised: Experimental Setup For robot obstacle avoidance tasks, the we use variants of the spiking actor network (SAN) [9] as host SNN.

Original: The original SAN uses LIF as its neuron model, but it resets themembrane potentials of all neurons to zero for each robot state.
Revised: The original SAN uses LIF as its neuron model, but it resets the membrane potentials of all neurons to zero for each robot state.

Original: The resting operation is contradictoryto the leaky function of LIF.
Revised: The resting operation is contradictory to the leaky function of LIF.

Original: Therefore, we modify the SAN by removing the resting operation, whichis dubbed SAN-NR.
Revised: Therefore, we modify the SAN by removing the resting operation, which is dubbed SAN-NR.

Original: To validate the effectiveness of the proposed BDETT, we integrate it into both
Revised: To validate the effectiveness of the proposed BDETT, we integrate it into both

Original: LIF-based and SRM-based SAN-NR models and compare them with their original static thresholdand two heuristic dynamic threshold schemes, DT1 [24] and DT2 [26].
Revised: LIF-based and SRM-based SAN-NR models and compare them with their original static threshold and two heuristic dynamic threshold schemes, DT1 [24] and DT2 [26].

Original: See Supplementary Note2 for details on the DT1 and DT2 schemes.
Revised: See Supplementary Note 2 for details on the DT1 and DT2 schemes.

Original: We set the batch size to 256 and the learning rate to0 .
Revised: We set the batch size to 256 and the learning rate to 0 .

Original: 00001 for both the actor and critic networks during the training process.
Revised: 00001 for both the actor and critic networks during the training process.

Original: In addition, we use the following hyperparameter settings for the proposed BDETT: η = 0 .
Revised: In addition, we use the following hyperparameter settings for the proposed BDETT: η = 0 .

Original: 01 and ψ = 4 . 

Revised: 01 and ψ = 4 .

Original: for the DET and
Revised: for the DET and

Original: C = 3 . 0 for the DTT.
Revised: C = 3 . 0 for the DTT.

Original: For estimating homeostasis, we set P = 200 .
Revised: For estimating homeostasis, we set P = 200 .

Original: See Supplementary Notes 4 forfurther training details.
Revised: See Supplementary Notes 4 for further training details.

Original: For robot continuous control, we adopt the population-coded SAN (PopSAN) [35] as our baselinemodel; it is a modified version of SAN [9] with a specifically designed encoder and decoder foraccommodating high-dimensional control tasks.
Revised: For robot continuous control, we adopt the population-coded SAN (PopSAN) [35] as our baseline model; it is a modified version of SAN [9] with a specifically designed encoder and decoder for accommodating high-dimensional control tasks.

Original: Note that PopSAN does not rest the membranepotentials as the encoder leverages soft-reset IF neurons.
Revised: Note that PopSAN does not rest the membrane potentials as the encoder leverages soft-reset IF neurons.

Original: Hence, PopSAN is the counterpart of the
Revised: Hence, PopSAN is the counterpart of the

Original: SAN-NR used in the obstacle avoidance tasks.
Revised: SAN-NR used in the obstacle avoidance tasks.

Original: We integrate BDETT into both LIF- and SRM-based
Revised: We integrate BDETT into both LIF- and SRM-based

Original: PopSAN models and compare them with their original static threshold schemes and the two heuristicdynamic schemes, DT1 and DT2.
Revised: PopSAN models and compare them with their original static threshold schemes and the two heuristic dynamic schemes, DT1 and DT2.

Original: Following the evaluation settings of PopSAN [35], we train tenmodels corresponding to ten random seeds, and the best-performing model is used for our assessmentconducted under different degraded conditions.
Revised: Following the evaluation settings of PopSAN [35], we train ten models corresponding to ten random seeds, and the best-performing model is used for our assessment conducted under different degraded conditions.

Original: In particular, the best-performing model is evaluatedten times under each experimental condition, and the mean reward of the ten evaluations represents themodel performance.
Revised: In particular, the best-performing model is evaluated ten times under each experimental condition, and the mean reward of the ten evaluations represents the model performance.

Original: Each evaluation consists of ten episodes, and each episode lasts for a maximumof 1000 execution steps.
Revised: Each evaluation consists of ten episodes, and each episode lasts for a maximum of 1000 execution steps.

Original: Hence, the P value used for estimating homeostasis is set to 100 , i .
Revised: Hence, the P value used for estimating homeostasis is set to 100 , i .

Original: e ., 10episodes × 10 evaluations.
Revised: e .,episodes × 10 evaluations.

Original: PopSAN and its variants are trained by using the twin-delayed deepdeterministic policy gradient off-policy algorithm [42].
Revised: PopSAN and its variants are trained by using the twin-delayed deep deterministic policy gradient off-policy algorithm [42].

Original: The hyperparameter settings of BDETT is thesame as the ones used for obstacle avoidance tasks, except the ψ for the DET is set to 6 . 0 .
Revised: The hyperparameter settings of BDETT is the same as the ones used for obstacle avoidance tasks, except the ψ for the DET is set to 6 . 0 .

Original: Followingthe training protocol of PopSAN [35], we set the batch size to 100 and the learning rate to 0 . {+
+}
Revised: Following the training protocol of PopSAN [35], we set the batch size to 100 and the learning rate to 0 . {+
+}

Original: 0001 forboth the actor and critic networks.
Revised: for both the actor and critic networks.

Original: The reward discount factor is set to 0 .
Revised: The reward discount factor is set to 0 .

Original: 99 , and the maximum lengthof the replay buffer is set to 1 million.
Revised: 99 , and the maximum length of the replay buffer is set to 1 million.

Original: See Supplementary Notes 5 formore training details related.
Revised: See Supplementary Notes 5 for training details.

Original: For the SRM-based baseline methods, the spike response kernel and refractory kernel of the SRMare adopted from [27, 29], and they are defined as ε ( t ) =
Revised: For the SRM-based baseline methods, the spike response kernel and refractory kernel of the SRM are adopted from [27, 29], and they are defined as ε ( t ) =

Original: te 1 − t and ζ ( t ) = − 2Θ( t ) e − t , respectively.
Revised: te 1 − t and ζ ( t ) = − 2Θ( t ) e − t , respectively.

Original: For all tasks, each dimension of a robot state is encoded into a spike train with T timesteps.
Revised: For all tasks, each dimension of a robot state is encoded into a spike train with T timesteps.

Original: Allexperimental results are obtained with T = 5 .
Revised: All experimental results are obtained with T = 5 .

Original: For a demonstration of the generalization provided bythe BDETT, we provide the experimental results obtained with T = 25 in Supplementary Notes 4, 5,6 for the obstacle avoidance, HalfCheetah-v3, and Ant-v3 tasks, respectively.
Revised: For a demonstration of the generalization provided by the BDETT, we provide the experimental results obtained with T = 25 in Supplementary Notes 4, 5, 6 for the obstacle avoidance, HalfCheetah-v3, and Ant-v3 tasks, respectively.

Original: 4.1 Robot Obstacle Avoidance with BDETT
Revised: 4.1 Robot Obstacle Avoidance with BDETT

Original: We evaluate the proposed method for robot obstacle avoidance tasks with one standard condition,i .
Revised: We evaluate the proposed method for robot obstacle avoidance tasks with one standard condition, i .

Original: e ., static obstacles, and three specifically designed adverse conditions: dynamic obstacles, degradedinputs, and weight uncertainty.
Revised: e ., static obstacles, and three specifically designed adverse conditions: dynamic obstacles, degraded inputs, and weight uncertainty.

Original: For the dynamic obstacle experiments, we introduce 11 dynamicallymoving cylinders in a static testing environment, and each repeatedly wanders between two points;see Figure 2a.
Revised: For the dynamic obstacle experiments, we introduce 11 dynamically moving cylinders in a static testing environment, and each repeatedly wanders between two points; see Figure 2a.

Original: The wandering distance and speed are designed to provide sufficient space and timeto allow possible passes.
Revised: The wandering distance and speed are designed to provide sufficient space and time to allow possible passes.

Original: The robot utilizes a Robo Peak light detection and ranging (RPLIDAR)system as its sensing device to detect obstacles, offering a field of view of 180 degrees with 18 rangemeasurements, as shown in Figure 2b.
Revised: The robot utilizes a Robo Peak light detection and ranging (RPLIDAR) system as its sensing device to detect obstacles, offering a field of view of 180 degrees with 18 range measurements, as shown in Figure 2b.

Original: In our degraded input scenario, we disturb the obtained range measurements in three different ways:“0.2": We set the range of the 3rd, 9th, and 15th lasers to 0.2 m, always reporting obstacles even whennone occur; “6.0":
Revised: In our degraded input scenario, we disturb the obtained range measurements in three different ways: “0.2": We set the range of the 3rd, 9th, and 15th lasers to 0.2 m, always reporting obstacles even when none occur; “6.0":

Original: This is similar to the “0.2" setting, but we set the three lasers’ ranges to 6.0 m,which is the average visible range in the test environment and means that the three lasers cannotperceive any obstacles; “GN": We add Gaussian noise [43] to each of the 18 range measurements.
Revised: This is similar to the “0.2" setting, but we set the three lasers’ ranges to 6.0 m, which is the average visible range in the test environment and means that the three lasers cannot perceive any obstacles; “GN": We add Gaussian noise [43] to each of the 18 range measurements.

Original: The three proposed degraded input settings are illustrated in Figure 2c.
Revised: The three proposed degraded input settings are illustrated in Figure 2c.

Original: In the weight uncertainty experiments, as illustrated in Figure 2d, the learned synaptic weightsof the host SNNs are also disturbed in three different ways.
Revised: In the weight uncertainty experiments, as illustrated in Figure 2d, the learned synaptic weights of the host SNNs are also disturbed in three different ways.

Original: “8-bit Loihi weight": Neuromorphichardware ( e . g ., Loihi) achieves computing efficiency by sacrificing the weight precision.
Revised: “8-bit Loihi weight": Neuromorphic hardware ( e . g ., Loihi) achieves computing efficiency by sacrificing the weight precision.

Original: Therefore,when deploying an SNN on neuromorphic hardware, one needs to scale and round up the learnedfloating-point synaptic weights to low-precision weights.
Revised: Therefore, when deploying an SNN on neuromorphic hardware, one needs to scale and round up the learned floating-point synaptic weights to low-precision weights.

Original: “GN weight": We add Gaussian noise toall weights.
Revised: “GN weight": We add Gaussian noise,

Revised: N (0 , 0 . 05) , to all synaptic weights.

Original: “30% zero weight": Among the synaptic weights between every two adjacent layers,we randomly set 30% of them to 0 .
Revised: “30% zero weight": Among the synaptic weights between every two adjacent layers, we randomly set 30% of them to 0 .

Original: To reduce the impact of the randomness introduced in the “GNweight" and “30% zero weight" experiments, we report the average success rates (SRs)  of 5-roundtests.
Revised: To reduce the impact of the randomness introduced in the “GN weight" and “30% zero weight" experiments, we report the average success rates (SRs) and standard deviation of 5-round tests.

Original: Success Rate
Revised: Success Rate

Original: The SRs of the competing LIF- and SRM-based approaches across all experimentalsettings are reported in Figure 2e.
Revised: The SRs of the competing LIF- and SRM-based approaches across all experimental settings are reported in Figure 2e and Table 1.

Original: For the “GN weight" and “30% zero weight" experiments, thestandard deviations of the 5-round SRs are also reported.
Revised: For the “GN weight" and “30% zero weight" experiments, the standard deviations of the 5-round SRs are also reported.

Original: The proposed BDETT achieves the highest SRs in all experiments, demonstrating its effectiveness.
Revised: The proposed BDETT achieves the highest SRs in all experiments, demonstrating its effectiveness.

Original: Notably, under dynamic obstacle conditions,the BDETT outperforms the runners-up by significant margins ( 9% versus the LIF and 12% versus the 
Revised: Notably, under dynamic obstacle conditions, the BDETT outperforms the runners-up by significant margins ( 9% versus the LIF and 12% versus the SRM).

Original: SRM degraded input conditions,  yields  least 10% more successful passes than 
Revised: Under degraded input conditions, the BDETT yields at least 10% more successful passes than other competing methods.

Original: In  experiments, ourincreases by least
Revised: In the weight uncertainty experiments, our

Revised: BDETT increases the SRs of the baseline SAN-NR model by at least 10 .

Original: % , and are also illustrated in Figure 2f
Revised: 5% , 24 . 6% , and 15 .

Original: (see Supplementary Figure 3 for results).
Revised: 6% under “8-bit Loihi weight", “GN weight", and “ 30% zero weight" settings, respectively.

Original: We  that our BDETT can help the robots effectively avoid both static and dynamic obstacles under all three adverse conditions; see 
Revised: We observe that our BDETT can help the robots effectively avoid both static and dynamic obstacles under all three adverse conditions; see Supplementary Tables 2,

Original: Supplementary Tables 2, and 4 for details. 
Revised: [Table] and 4 for details. [Table] [Table]

Original: Homeostatic Evaluation When an SNN is in homeostasis, all neurons are expected to have similarand sparse firing patterns under different conditions [44, 45].
Revised: Homeostatic Evaluation When an SNN is in homeostasis, all neurons are expected to have similar and sparse firing patterns under different conditions [44, 45].

Original: Therefore, when transferring from onecondition to another, the SNNs with stronger homeostasis are expected to induce fewer changes inall three metrics.
Revised: Therefore, when transferring from one condition to another, the SNNs with stronger homeostasis are expected to induce fewer changes in all three metrics.

Original: The changes induced in all successful trials involving the LIF- and SRM-basedhost SNNs under different experimental settings are illustrated in Figures 2g and h, respectively.
Revised: The changes induced in all successful trials involving the LIF- and SRM-based host SNNs under different experimental settings are illustrated in Figures 2f and g, respectively [Table]

Original: The changes ( i .
Revised: The changes ( i .

Original: e ., in ∆ FR m , ∆ FR mstd , and ∆ FR sstd ) are estimated with respect to the correspondinghomeostasis achieved in the dynamic obstacle experiments, i .
Revised: e ., in ∆ FR m , ∆ FR mstd , and ∆ FR sstd ) are estimated with respect to the corresponding homeostasis achieved in the dynamic obstacle experiments, i [Table]

Original: e ., under the base condition.
Revised: e ., under the base condition.

Original: Theproposed BDETT scheme yields minimal changes in all three metrics when transferring from the basecondition to all other experimental settings, except for the ∆ FR s std estimated based on the SRM-based8 -bit Loihi weight experiment.
Revised: The proposed BDETT scheme yields minimal changes in all three metrics when transferring from the base condition to all other experimental settings, except for the ∆ FR sstd estimated based on the SRM-based 8 -bit Loihi weight experiment.

Original: The figures highlight that the proposed BDETT significantly improveson the baseline SAN-NR model, as evidenced by the remarkable drops in these three statisticalmetrics.
Revised: The figures highlight that the proposed BDETT significantly improves on the baseline SAN-NR model, as evidenced by the remarkable drops in these three statistical metrics.

Original: For example, as shown in the “6.0" section of the ∆ FR m in Figure 2g, our dynamic thresholdscheme reduces the ∆ FR m from 0 .
Revised: For example, as shown in the “6.0" section of the ∆ FR m in Figure 2f, our dynamic threshold scheme reduces the ∆ FR m from 0 .

Original: 043 to 0 .
Revised: 043 to 0 .

Original: In the “0.2" section of the ∆ FR sstd in Figure 2h, the∆ FR sstd is decreased to 1 .
Revised: In the “0.2" section of the ∆ FR sstd in Figure 2g, the ∆ FR sstd is decreased to 1 .

Original: 7% of its original value (from 0 .
Revised: 7% of its original value (from 0 .

Original: 0058 to 0 . 0001 ).
Revised: 0058 to 0 . 0001 ).

Original: We also witness that the
Revised: We also witness that the

Original: The LIF- and SRM-based SNNs’ homeostasis changes with respect to the ‘Base’ condition in the

Original: HalfCheetah-v3 tasks.

Original: h & i. The LIF- and SRM-based SNNs’ homeostasis changes with respect to the ‘Base’ condition in the Ant-v3 tasks.

Original: d-i use the same color codes shown in e

Original: DT1 and DT2 schemes significantly weaken the baseline model’s homeostasis, as shown in Figure 2gin the “0.2" section of the ∆ FR mstd and the “6.0" section of the ∆ FR sstd .
Revised: DT1 and DT2 schemes significantly weaken the baseline model’s homeostasis, as shown in Figure 2f in the “0.2" section of the ∆ FR mstd and the “6.0" section of the ∆ FR sstd .

Original: The goal of homeostasis to enhance the host SNN generalization.
Revised: The goal of homeostasis to enhance the host SNN generalization.

Original: Therefore, we expect SNNs withstronger homeostasis ( i .
Revised: Therefore, we expect SNNs with stronger homeostasis ( i .

Original: e ., smaller ∆ FR m , ∆ FR mstd , and ∆ FR sstd values) to outperform those withweaker homeostasis.
Revised: e ., smaller ∆ FR m , ∆ FR mstd , and ∆ FR sstd values) to outperform those with weaker homeostasis.

Original: Our experimental results confirm this, validating that the strong homeostasisprovided by our BDETT can improve the generalization capabilities of SNNs to different degradedconditions.
Revised: Our experimental results confirm this, validating that the strong homeostasis provided by our BDETT can improve the generalization capabilities of SNNs to different degraded The LIF SRM-based SNNs’ homeostasis with respect to the ‘Base’ condition in the SNNs’ conditions.

Original: We argue that this is a highly desired capability not only for mobile robotics but also forbroader machine learning.
Revised: We argue that this is a highly desired capability not only for mobile robotics but also for broader machine learning.

Original: See Supplementary Note 4 for more experimental results and analysis.
Revised: See Supplementary Note 4 for more experimental results and analysis.

Original: 4.2 Continuous Robot Control with BDETT
Revised: 4.2 Continuous Robot Control with BDETT

Original: For the HalfCheetah-v3 and Ant-v3 tasks, similar to the robot obstacle avoidance tasks, we evaluate onone standard and two specifically designed degraded inputs and weight uncertainty adverse conditionsto demonstrate the strong generalization enabled by our BDETT.
Revised: For the HalfCheetah-v3 and Ant-v3 tasks, similar to the robot obstacle avoidance tasks, we evaluate on one standard and two specifically designed degraded inputs and weight uncertainty adverse conditions to demonstrate the strong generalization enabled by our BDETT.

Original: In this context, for the degradedinput conditions, we disturb the observations of these two control tasks in three ways.
Revised: In this context, for the degraded input conditions, we disturb the observations of these two control tasks in three ways.

Original: “Random jointposition": For each episode, one of the joint positions is randomly selected, and its original positionis replaced by a random number sampled from a Gaussian distribution N (0 , 0 . 1) .
Revised: “Random joint position": For each episode, one of the joint positions is randomly selected, and its original position is replaced by a random number sampled from a Gaussian distribution N (0 , 0 . 1) .

Original: “Random jointvelocity": We randomly select one of the joint velocities in each episode and change its observedvelocity to a random number sampled from a Gaussian distribution N (0 , 10 . 0) .
Revised: “Random joint velocity": We randomly select one of the joint velocities in each episode and change its observed velocity to a random number sampled from a Gaussian distribution N (0 , 10 . 0) .

Original: “GN": In eachepisode, we add Gaussian noise sampled from the distribution N (0 , 1 . 0) to each dimension of a state;see Figure 3b.
Revised: “GN": In each episode, we add Gaussian noise sampled from the distribution N (0 , 1 . 0) to each dimension of a state; see Figure 3b.

Original: The weight uncertainty conditions of the control tasks are the same as those used inthe robot obstacle avoidance tasks, as illustrated in Figure 3c.
Revised: The weight uncertainty conditions of the control tasks are the same as those used in the robot obstacle avoidance tasks, as illustrated in Figure 3c.

Original: Rewards As shown in Figures 3d and e, under all experimental settings, the proposed BDETToffers the host SNNs the highest rewards, significantly improving upon the rewards of the baseline PopSAN model by at least 438 ( i .
Revised: Rewards As shown in Figures 3d, e and Table 2, under all experimental settings, the proposed BDETT offers the host SNNs the highest rewards, significantly improving upon the rewards of the baseline PopSAN model by at least 438 ( i .

Original: e ., the SRM-based PopSAN model under the “GN" setting) for the HalfCheetah-v3 tasks and 213 ( i .
Revised: e ., the SRM-based PopSAN model under the “GN" setting) for the HalfCheetah-v3 tasks and 213 ( i .

Original: e ., the LIF-based PopSAN model under the “Random joint velocity"setting) for the Ant-v3 tasks.
Revised: e ., the LIF-based PopSAN model under the “Random joint velocity" setting) for the Ant-v3 tasks.

Original: Notably, under weight uncertainty conditions with a HalfCheetah-v3agent, even with low-precision 8-bit weights, the proposed BDETT helps the SRM-based host SNNachieve a higher reward than that obtained with high-precision floating-point weights (11767 vs.11268); see Supplementary Tables 7 and 9.
Revised: Notably, under weight uncertainty conditions with a HalfCheetah-v3 agent, even with low-precision 8-bit weights, the proposed BDETT helps the SRMbased host SNN achieve a higher reward than that obtained with high-precision floating-point weights (11767 vs. 11268); see Supplementary Tables 7 and 9.

Original: With an Ant-v3 agent, the proposed BDETT helps boththe LIF- and SRM-based host SNNs achieve higher rewards, even with low-precision weights, i .
Revised: With an Ant-v3 agent, the proposed BDETT helps both the LIF- and SRM-based host SNNs achieve higher rewards, even with low-precision weights, i .

Original: e .,5570 vs. 5526 and 5648 vs. 5643, respectively.
Revised: e ., 5570 vs. 5526 and 5648 vs. 5643, respectively.

Original: See Supplementary Notes 5 and 6 for additionalexperimental results and analysis related to the HalfCheetah-v3 and Ant-v3 tasks, respectively.
Revised: See Supplementary Notes 5 and estimated [Table] for additional experimental results and analysis related to the HalfCheetah-v3 and Ant-v3 tasks, respectively.

Original: Homeostatic Evaluation In Figures 3f-i, we show the changes induced in these three metrics whenshifting from normal conditions ( i .
Revised: Homeostatic Evaluation In Figures 3f-i, we show the changes induced in these three metrics when shifting from normal conditions ( i .

Original: e ., the base conditions) to all other experimental settings.
Revised: e ., the base conditions) to all other experimental settings.

Original: Theproposed BDETT offers the strongest homeostasis to the host SNNs among all competing approachesfor both the HalfCheetah-v3 and Ant-v3 control tasks.
Revised: The proposed BDETT offers the strongest homeostasis to the host SNNs among all competing approaches for both the HalfCheetah-v3 and Ant-v3 control tasks.

Original: In particular, for the HalfCheetah-v3 control task, as shown in the “30% zero weight" section ofthe ∆ FR m in Figure 3f, our dynamic threshold scheme reduces the ∆ FR m of the baseline PopSANmodel from 0 .
Revised: In particular, for the HalfCheetah-v3 control task, as shown in the “30% zero weight" section of the ∆ FR m in Figure 3f, our dynamic threshold scheme reduces the ∆ FR m of the baseline PopSAN model from 0 .

Original: 069 to 0 .
Revised: 069 to 0 .

Original: In the “GN weight" section of the ∆ FR sstd in Figure 3g, the proposed BDETT decreases the ∆ FR sstd of the SRM-based PopSAN to 8 .
Revised: In the “GN weight" section of the ∆ FR sstd in Figure 3g, the proposed BDETT decreases the ∆ FR sstd of the SRM-based PopSAN to 8 .

Original: 3% of its original value (from 0 .
Revised: 3% of its original value (from 0 .

Original: 0012to 0 . 0001 )
Revised: 0012 to 0 . 0001 )

Original: ; see Supplementary Table 10 for details.
Revised: ; see Supplementary Table 10 for details.

Original: For the Ant-v3 control task, as shown in the“GN weight" section of the ∆ FR m in Figure 3h, our dynamic threshold scheme reduces the ∆ FR mof the LIF-based baseline model from 0  .
Revised: For the Ant-v3 control task, as shown in the “GN weight" section of the ∆ FR m in Figure 3h, our dynamic threshold scheme reduces the ∆ FR m of the LIF-based baseline model from 0 .to 0 .

Original: 041 to 0 .

Original: In the “Random joint position" section ofthe ∆ FR sstd in Figure 3h, the ∆ FR sstd of the LIF-based baseline model is decreased to 10% of itsoriginal value (from 0 . 0010 to 0 . 0001 ); see Supplementary Table 15 for details.
Revised: In the “Random joint position" section of the ∆ FR sstd in Figure 3h, the ∆ FR sstd of the LIF-based baseline model is decreased to 10% of its original value (from 0 . 0010 to 0 . 0001 ); see Supplementary Table 15 for details.

Original: As in the obstacleavoidance tasks, the DT1 and DT2 schemes significantly decrease the homeostasis of both the LIFand SRM-based baseline models in both continuous control tasks.
Revised: As in the obstacle avoidance tasks, the DT1 and DT2 schemes significantly decrease the homeostasis of both the LIF- and SRM-based baseline models in both continuous control tasks.

Original: Some extreme cases are shown inthe “Random joint velocity" section of the ∆ FR mstd in Figure 3f, and the “GN weight" section of the∆ FR m in Figure 3i.
Revised: Some extreme cases are shown in the “Random joint velocity" section of the ∆ FR mstd in Figure 3f, and the “GN weight" section of the ∆ FR m in Figure 3i.

Original: These experimental results obtained for the two continuous control tasks support the observationsobtained in the obstacle avoidance tasks.
Revised: These experimental results obtained for the two continuous control tasks support the observations obtained in the obstacle avoidance tasks.

Original: More importantly, we witness that the strong homeostasisprovided by our BDETT improves generalization to severely degraded conditions.
Revised: More importantly, we witness that the strong homeostasis provided by our BDETT improves generalization to severely degraded conditions.

Original: 4.3  BDETTwithout Statistical Parameter Settings
Revised: 4.3 Image Classification with BDETT

Revised: We assess the proposed SNN-based learning method on image classification as a relevant vision task.

Revised: In particular, to measure the generalization of the proposed BDETTscheme, we conduct additional image classification experiments under normal and degraded conditions.

Revised: To this end, we simulate degraded inputs similar to the robotic control tasks; see the weight uncertainty degraded settings illustrated in Figure 4d.

Revised: In addition, we test on degradations that are tailored to classification from two adversarial attack methods; the fast gradient sign method (FGSM) [46] and projected gradient descent (PGD) [47]; see Figure 4c.

Revised: Specifically, we train the SCNN model [48] on the MNIST dataset [49] as our baseline model.

Revised: Each pixel of an MNIST image is encoded into 30 Poisson spikes as inputs to SCNN for training and testing.

Revised: As shown in Figure 4e and Table 3, directly applying the proposed approach without any changes to image classification in degraded conditions compares favorably across all experimental settings and in terms of generalization.

Revised: For stronger degradations, the Top-1 classification accuracy of both the baseline and our approach decreases, but the proposed method is less affected.

Revised: Note that the SCNN model contains CNN layers, blocking us from estimating homeostasis.

Revised: 4.4 BDETT without Statistical Parameter Adjustment

Original: We found it essentialfor the proposed threshold scheme to replace the constants in the two originalbiological models  with layerwise statistical cues.
Revised: We found it essential to replace the constants in the two biological models we base our approach on with layerwise statistical cues.

Original: Here, we report the performance of the BDETTwith the original constants of the fitted biological models, demonstrating the effectiveness of theproposed layerwise statistical parameter settings.
Revised: Here, we report the performance of the BDETT with the original constants of the fitted biological models, demonstrating the effectiveness of the proposed layerwise statistical parameter settings.

Original: In particular, we adopt the corresponding constantsof the fitted adaptive threshold model [16] and replaced the V lm ( t ) and V lθ ( t ) , i .
Revised: In particular, we first use the corresponding constants of the fitted adaptive threshold model [16] and replaced the V lm ( t ) and V lθ ( t ) , i .

Original: e .
Revised: e .

Original: , Eq.
Revised: , Eq.

Original: 3 and Eq.
Revised: 3 and Eq.

Original: 4,with 3 and 7 , respectively.
Revised: 4, with 3 and 7 , respectively.

Original: These two constants are obtained by shifting the originally fitted constants− 67 and − 63 by 70 to compensate for the difference of the rest potentials; − 70 mV in the originalmodel but 0 mV  for LIF and SRM models.
Revised: These two constants are obtained by shifting the originally fitted constants − 67 and −by 70 to compensate for the difference of the rest potentials; − 70 mV in the original model but 0 mV [Table] for LIF and SRM models.

Original: Furthermore, we use the original fitted parameters in our DTT, and Eq.
Revised: Furthermore, we use the original fitted parameters in our DTT, and Eq.becomes

Original: [-5 becomes T li ( t + 1) = 1 . 
 + 10 e − ( vli ( t +1) − vli ( t )).-]
Revised: [-5 becomes T li ( t + 1) = 1 . 
 + 10 e − ( vli ( t +1) − vli ( t )).-][Equation]

Original: Forthe obstacle avoidance task, withthe originally fitted constants, the LIF-based policy cannot produce any successful pass even underthe standard testing condition; SR drops from 92.5% to 0%.
Revised: For obstacle avoidance, with the originally fitted constants, the LIF-based policy cannot produce any successful pass even under the standard testing condition; SR drops from 92.5% to 0%.

Original: For the HalfCheetah-v3 and Ant-v3 tasksunder the standard testing condition, with the originally fitted constants, the rewards achieved by a LIF-based policy dropped from 11064 to − 35 and 5276 to − 9 , respectively.
Revised: For the HalfCheetah-v3 and Ant-v3 tasks, with the originally fitted constants, the rewards achieved by a LIF-based policy dropped from 11064 to − 35 andto − 9 , respectively.

Original: Note that an untrained BDETT-based policy achieves − 124 and − 73 rewards for these two continuous control tasks.
Revised: Note that an untrained BDETT-based policy achieves − 124 and − 73 rewards for these two continuous control tasks.

Revised: Image classification tasks follow the same pattern; that is, with the originally fitted constants, the Top-1 accuracy achieved by a LIF-based policy dropped from 99.45% to 9.80%.

Original: Theseexperimental results validate that the proposed statistical cues are essential to the proposed method.
Revised: These experimental results validate that the proposed statistical cues are essential to the proposed method.

Original: See Supplementary Note 9 for more experimental results.

Original:  Conclusion
Revised:  Conclusion

Original: This work introduces a novel biologically inspired BDETT scheme to SNNs that significantly improves generalization, and as such, fills a gap between biological research and machine learning.
Revised: This work introduces a novel biologically inspired BDETT scheme to SNNs that significantly improves generalization, and as such, fills a gap between biological research and machine learning.

Original: Dynamic threshold behavior plays an essential role in maintaining a neuronal homeostasis in biological nervous systems.
Revised: Dynamic threshold behavior plays an essential role in maintaining a neuronal homeostasis in biological nervous systems.

Original: Motivated by this observation, we propose a dynamic threshold scheme toachieve homeostasis in artificial SNNs.
Revised: Motivated by this observation, we propose a dynamic threshold scheme to achieve homeostasis in artificial SNNs.

Original: We assess the proposed approach in real-world tasks undernormal and severely degraded conditions to validate its generalization capabilities.
Revised: We assess the proposed approach in real-world tasks under normal and severely degraded conditions to validate its generalization capabilities.

Original: We find thatthe proposed dynamic threshold achieves strong homeostasis along with generalization to diversedegraded conditions.
Revised: We find that the proposed dynamic threshold achieves strong homeostasis along with generalization to diverse degraded conditions.

Original: This finding is a step toward employing bioplausible SNNs in real-worldapplications.
Revised: This finding is a step toward employing bioplausible SNNs in real-world applications.

Original: As future work, we plan to implement the proposed scheme on neuromorphic hardwareto broadly deploy BDETT in future robotic platforms. 
Revised: As future work, we plan to implement the proposed scheme on neuromorphic hardware to broadly deploy BDETT in future robotic platforms. Acknowledgements

Revised: This work was supported in part by National Key Research and Development Program of China (2022ZD0210500, 2018AAA0102003, 2021ZD0112400), the National Natural Science Foundation of China under Grant 61972067/U21A20491/U1908214, and the Innovation Technology Funding of

Revised: Dalian (2020JJ26GX036).

Revised: Felix Heide was supported by an NSF CAREER Award (2047359), a Sony

Revised: Young Faculty Award, a Project X Innovation Award, and an Amazon Science Research Award.

Original: Checklist
Revised: Checklist

Original: For all authors... a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
Revised: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contribu- tions and scope?

Original: [Yes]( b) Did you describe the limitations of your work?
Revised: [Yes] (b) Did you describe the limitations of your work?

Original: [Yes] The limitations are pointed out at the endof introduction.
Revised: [Yes] The limitations are pointed out at the end of introduction.

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [N/A] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
Revised: [N/A] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?

Original: [Yes] 
Revised: [Yes] 2.

Original: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?
Revised: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?

Original: [N/A] (b) Did you include complete proofs of all theoretical results?
Revised: [N/A] (b) Did you include complete proofs of all theoretical results?

Original: [N/A] 
Revised: [N/A] 3.

Original: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimentalresults (either in the supplemental material or as a URL)?
Revised: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

Original: [No] We will release the code upon acceptance.
Revised: [No] We will release the code upon acceptance.

Original: (b) Did you specify all the training details (e g., data splits, hyperparameters, how they were chosen)?
Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Original: [Yes] See section 4.1, Supplementary note 4 and 5.
Revised: [Yes] See section 4.1, Supplementary note 4 and 5.

Original: (c) Did you report error bars (e g., with respect to the random seed after running experiments multiple times)?
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Original: [Yes] (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs,internal cluster, or cloud provider)?
Revised: [Yes] (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?

Original: [Yes] See supplementary note 4 and 5.
Revised: [Yes] See supplementary note 4 and 5.

Original: If you are using existing assets (e g., code, data, models) or curating/releasing new assets...
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...

Original: (a) If your work uses existing assets, did you cite the creators?
Revised: (a) If your work uses existing assets, did you cite the creators?

Original: [Yes] (b) Did you mention the license of the assets?
Revised: [Yes] (b) Did you mention the license of the assets?

Original: [N/A] (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: [N/A] (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [N/A]
Revised: [N/A]

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re us- ing/curating?
Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re us- ing/curating?

Original: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable informa- tion or offensive content?
Revised: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable informa- tion or offensive content?

Original: [N [Table]
Revised: [N/A] 5.

Original: [Table]
Revised: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: [Table]
Revised: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?

Original: [Table] [Table] [Table] [Table]
Revised: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?

Original: [Table]
Revised: [N/A]