Original: E FFICIENT I NFERENCE OF F LEXIBLE I NTERACTION IN S PIKING - NEURON N ETWORKS
Revised: E FFICIENT I NFERENCE OF F LEXIBLE I NTERACTION IN S PIKING - NEURON N ETWORKS

Original: A BSTRACT
Revised: A BSTRACT

Original: Hawkes process provides an effective statistical framework for analyzing the timedependent interaction of neuronal spiking activities.
Revised: Hawkes process provides an effective statistical framework for analyzing the timedependent interaction of neuronal spiking activities.

Original: Although utilized in many real applications, the classic Hawkes process is incapable of modelling inhibitory interactions among neurons.
Revised: Although utilized in many real applications, the classic Hawkes process is incapable of modelling inhibitory interactions among neurons.

Original: Instead, the nonlinear Hawkes process allows for a more ﬂexible inﬂuence pattern with excitatory or inhibitory interactions.
Revised: Instead, the nonlinear Hawkes process allows for a more ﬂexible inﬂuence pattern with excitatory or inhibitory interactions.

Original: In this paper, three sets of auxiliary latent variables (P ´ olya-Gamma variables, latent marked Poisson processes and sparsity variables) are augmented to make functional connection weights in a Gaussian form which allows for a simple iterative algorithm with analytical updates.
Revised: In this paper, three sets of auxiliary latent variables (P ´ olya-Gamma variables, latent marked Poisson processes and sparsity variables) are augmented to make functional connection weights in a Gaussian form, which allows for a simple iterative algorithm with analytical updates.

Original: As a result, an efﬁcient expectationmaximization (EM) algorithm is derived to obtain the maximum a posteriori (MAP) estimate.
Revised: As a result, an efﬁcient expectationmaximization (EM) algorithm is derived to obtain the maximum a posteriori (MAP) estimate.

Original: We demonstrate the accuracy and efﬁciency performance of our algorithm on synthetic and real data.
Revised: We demonstrate the accuracy and efﬁciency performance of our algorithm on synthetic and real data.

Original: For real neural recordings, we show our algorithm can estimate the temporal dynamics of interaction and reveal the interpretable functional connectivity underlying neural spike trains
Revised: For real neural recordings, we show our algorithm can estimate the temporal dynamics of interaction and reveal the interpretable functional connectivity underlying neural spike trains

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: One of the most important tracks in neuroscience is to examine the neuronal activity in the cerebral cortex under varying experimental conditions.
Revised: One of the most important tracks in neuroscience is to examine the neuronal activity in the cerebral cortex under varying experimental conditions.

Original: Recordings of neuronal activity are represented through a series of action potentials or spike trains.
Revised: Recordings of neuronal activity are represented through a series of action potentials or spike trains.

Original: The transmitted information and functional connection between neurons are considered to be primarily represented by spike trains (Kass et al., 2014; Kass & Ventura, 2001; Brown et al., 2004; 2002).
Revised: The transmitted information and functional connection between neurons are considered to be primarily represented by spike trains (Kass et al., 2014; Kass & Ventura, 2001; Brown et al., 2004; 2002).

Original: A spike train is a sequence of recorded times at which a neuron ﬁres an action potential and each spike may be considered to be a timestamp.
Revised: A spike train is a sequence of recorded times at which a neuron ﬁres an action potential and each spike may be considered to be a timestamp.

Original: Spikes occur irregularly both within and across multiple trials, so it is reasonable to consider a spike train as a point process with the instantaneous ﬁring rate being the intensity function of point processes (Perkel et al., 1967; Paninski, 2004; Eden et al., 2004).
Revised: Spikes occur irregularly both within and across multiple trials, so it is reasonable to consider a spike train as a point process with the instantaneous ﬁring rate being the intensity function of point processes (Perkel et al., 1967; Paninski, 2004; Eden et al., 2004).

Original: An example of spike trains for multiple neurons is shown in Fig.
Revised: An example of spike trains for multiple neurons is shown in Fig.

Original: 2a in the real data experiment.
Revised: 2a in the real data experiment.

Original: Despite many existing applications, the classic point process models, e.g., Poisson processes, neglect the time-dependent interaction within one neuron and between multiple neurons, so fail to capture the complex temporal dynamics of a neural population.
Revised: Despite many existing applications, the classic point process models, e.g., Poisson processes, neglect the time-dependent interaction within one neuron and between multiple neurons, so fail to capture the complex temporal dynamics of a neural population.

Original: In contrast, Hawkes process is one type of point processes which is able to model the self-exciting interaction between past and future events.
Revised: In contrast, Hawkes process is one type of point processes which is able to model the self-exciting interaction between past and future events.

Original: Existing applications cover a wide range of domains including seismology (Ogata, 1998; 1999), criminology (Mohler et al., 2011; Lewis et al., 2012), ﬁnancial engineering (Bacry et al., 2015; Filimonov & Sornette, 2015) and epidemics (Saichev & Sornette, 2011; Rizoiu et al., 2018).
Revised: Existing applications cover a wide range of domains including seismology (Ogata, 1998; 1999), criminology (Mohler et al., 2011; Lewis et al., 2012), ﬁnancial engineering (Bacry et al., 2015; Filimonov & Sornette, 2015) and epidemics (Saichev & Sornette, 2011; Rizoiu et al., 2018).

Original: Unfortunately, due to the linearly additive intensity, the vanilla Hawkes process can only represent the purely excitatory interaction because a negative ﬁring rate may exist with inhibitory interaction.
Revised: Unfortunately, due to the linearly additive intensity, the vanilla Hawkes process can only represent the purely excitatory interaction because a negative ﬁring rate may exist with inhibitory interaction.

Original: This makes the vanilla version inappropriate in the neuroscience domain where the inﬂuence between neurons is a mixture of excitation and inhibition (Maffei et al., 2004; Mongillo et al., 2018).
Revised: This makes the vanilla version inappropriate in the neuroscience domain where the inﬂuence between neurons is a mixture of excitation and inhibition (Maffei et al., 2004; Mongillo et al., 2018).

Original: In order to reconcile Hawkes process with inhibition, various nonlinear Hawkes process variants are proposed to allow for both excitatory and inhibitory interactions.
Revised: In order to reconcile Hawkes process with inhibition, various nonlinear Hawkes process variants are proposed to allow for both excitatory and inhibitory interactions.

Original: The core point of nonlinear
Revised: The core point of nonlinear

Original: Hawkes process is a nonlinearity which maps the convolution of the spike train with a causal inﬂuential kernel to a nonnegative conditional intensity, such as rectiﬁer (Reynaud-Bouret et al., 2013), exponential (Gerhard et al., 2017) and sigmoid (Linderman, 2016; Apostolopoulou et al., 2019).
Revised: Hawkes process is a nonlinearity which maps the convolution of the spike train with a causal inﬂuential kernel to a nonnegative conditional intensity, such as rectiﬁer (Reynaud-Bouret et al., 2013), exponential (Gerhard et al., 2017) and sigmoid (Linderman, 2016; Apostolopoulou et al., 2019).

Original: The sigmoid mapping function has the advantage that the P´olya-Gamma augmentation scheme can be utilized to convert the likelihood into a Gaussian form, which makes the inference tractable.
Revised: The sigmoid mapping function has the advantage that the P´olya-Gamma augmentation scheme can be utilized to convert the likelihood into a Gaussian form, which makes the inference tractable.

Original: In Linderman (2016), a discrete-time model is proposed to convert the likelihood from a Poisson process to a Poisson distribution.
Revised: In Linderman (2016), a discrete-time model is proposed to convert the likelihood from a Poisson process to a Poisson distribution.

Original: Then P´olya-Gamma random variables are augmented on discrete observations to propose a Gibbs sampler.
Revised: Then P´olya-Gamma random variables are augmented on discrete observations to propose a Gibbs sampler.

Original: This method is further extended to a continuous-time regime in Apostolopoulou et al.
Revised: This method is further extended to a continuous-time regime in Apostolopoulou et al.

Original: by augmenting thinned points and P´olya-Gamma random variables to propose a Gibbs sampler.
Revised: by augmenting thinned points and P´olya-Gamma random variables to propose a Gibbs sampler.

Original: However, the inﬂuence function is limited to be purely exciting or inhibitive exponential decay.
Revised: However, the inﬂuence function is limited to be purely exciting or inhibitive exponential decay.

Original: Besides, due to the nonconjugacy of the excitation parameter of exponential decay inﬂuence function, a Metropolis-Hastings sampling step has to be embedded into the
Revised: Besides, due to the nonconjugacy of the excitation parameter of exponential decay inﬂuence function, a Metropolis-Hastings sampling step has to be embedded into the

Original: Gibbs sampler making the Markov chain Monte Carlo (MCMC) algorithm further inefﬁcient.
Revised: Gibbs sampler making the Markov chain Monte Carlo (MCMC) algorithm further inefﬁcient.

Original: To address the parametric and inefﬁcient problems in aforementioned existing works, we develop a ﬂexible sigmoid nonlinear multivariate Hawkes processes (SNMHP) model in the continuous-time regime, (1) which can represent the ﬂexible excitation-inhibition-mixture temporal dynamics among the neural population, (2) with the efﬁcient conjugate inference.
Revised: To address the parametric and inefﬁcient problems in aforementioned existing works, we develop a ﬂexible sigmoid nonlinear multivariate Hawkes processes (SNMHP) model in the continuous-time regime, (1) which can represent the ﬂexible excitation-inhibition-mixture temporal dynamics among the neural population, (2) with the efﬁcient conjugate inference.

Original: An EM inference algorithm is proposed to ﬁt neural spike trains.
Revised: An EM inference algorithm is proposed to ﬁt neural spike trains.

Original: Inspired by Donner & Opper (2017; 2018), three auxiliary latent variable sets: P ´ olya-Gamma variables, latent marked Poisson processes and sparsity variables are augmented to make functional connection weights in a Gaussian form.
Revised: Inspired by Donner & Opper (2017; 2018), three auxiliary latent variable sets: P´olya-Gamma variables, latent marked Poisson processes and sparsity variables are augmented to make functional connection weights in a Gaussian form.

Original: As a result, the EM algorithm has analytical updates with drastically improved efﬁciency.
Revised: As a result, the EM algorithm has analytical updates with drastically improved efﬁciency.

Original: As shown in experiments, it is even more efﬁcient than the maximum likelihood estimation (MLE) for the parametric Hawkes process in high dimensional cases.
Revised: As shown in experiments, it is even more efﬁcient than the maximum likelihood estimation (MLE) for the parametric Hawkes process in high dimensional cases.

Original:  O UR M ODEL
Revised:  O UR M ODEL

Original: Neurons communicate with each other by action potentials (spikes) and chemical neurotransmitters.
Revised: Neurons communicate with each other by action potentials (spikes) and chemical neurotransmitters.

Original: A spike causes the pre-synaptic neuron to release a chemical neurotransmitter that induces impulse responses, either exciting or inhibiting the post-synaptic neuron from ﬁring its own spikes.
Revised: A spike causes the pre-synaptic neuron to release a chemical neurotransmitter that induces impulse responses, either exciting or inhibiting the post-synaptic neuron from ﬁring its own spikes.

Original: The addition of excitatory and inhibitory inﬂuence to a neuron determines whether a spike will occur.
Revised: The addition of excitatory and inhibitory inﬂuence to a neuron determines whether a spike will occur.

Original: At the same time, the impulse response characterizes the temporal dynamics of the exciting or inhibiting inﬂuence which can be complex and ﬂexible (Purves et al., 2014; Squire et al., 2012; Bassett & Sporns, 2017).
Revised: At the same time, the impulse response characterizes the temporal dynamics of the exciting or inhibiting inﬂuence which can be complex and ﬂexible (Purves et al., 2014; Squire et al., 2012; Bassett & Sporns, 2017).

Original: Arguably, the ﬂexible nonlinear multivariate Hawkes processes are a suitable choice for representing the temporal dynamics of mutually excitatory or inhibitory interactions and functional connectivity of neuron networks.
Revised: Arguably, the ﬂexible nonlinear multivariate Hawkes processes are a suitable choice for representing the temporal dynamics of mutually excitatory or inhibitory interactions and functional connectivity of neuron networks.

Original: 2. M ULTIVARIATE H AWKES P ROCESSES
Revised: 2. M ULTIVARIATE H AWKES P ROCESSES

Original: The vanilla multivariate Hawkes processes (Hawkes, 1971) are sequences of timestamps D
Revised: The vanilla multivariate Hawkes processes (Hawkes, 1971) are sequences of timestamps D

Original: = {{ t in } N i n =1 } Mi =1 ∈ [0 , T ] where t in is the timestamp of n -th event on i -th dimension with N i being the number of points on i -th dimension, M the number of dimensions, T the observation window.
Revised: = {{ t in } N i n =1 } Mi =1 ∈ [0 , T ] where t in is the timestamp of n -th event on i -th dimension with N i being the number of points on i -th dimension, M the number of dimensions, T the observation window.

Original: The i -th dimensional conditional intensity, the probability of an event occurring on i -th dimension in [ t, t + dt ) given all dimensional history before t , is designed in a linear superposition form:
Revised: The i -th dimensional conditional intensity, the probability of an event occurring on i -th dimension in [ t, t + dt ) given all dimensional history before t , is designed in a linear superposition form:

Original: [Equation]
Revised: [Equation]

Original: where µ i > 0 is the baseline rate of i -th dimension and φ ij ( · ) ≥ 0 is the causal inﬂuence function (impulse response) from j -th dimension to i -th dimension which is normally a parameterized function, e.g., exponential decay.
Revised: where µ i > 0 is the baseline rate of i -th dimension and φ ij ( · ) ≥ 0 is the causal inﬂuence function (impulse response) from j -th dimension to i -th dimension which is normally a parameterized function, e.g., exponential decay.

Original: The summation explains the self- and mutual-excitation phenomenon, i.e., the occurrence of previous events increases the intensity of events in the future.
Revised: The summation explains the self- and mutual-excitation phenomenon, i.e., the occurrence of previous events increases the intensity of events in the future.

Original: Unfortunately, one blemish is the vanilla multivariate Hawkes processes allow only nonnegative (excitatory) inﬂuence functions because negative (inhibitory) inﬂuence functions may yield a negative intensity which is meaningless.
Revised: Unfortunately, one blemish is the vanilla multivariate Hawkes processes allow only nonnegative (excitatory) inﬂuence functions because negative (inhibitory) inﬂuence functions may yield a negative intensity which is meaningless.

Original: To reconcile the vanilla version with inhibitory effect and ﬂexible inﬂuence function, we propose the SNMHP.
Revised: To reconcile the vanilla version with inhibitory effect and ﬂexible inﬂuence function, we propose the SNMHP.

Original: 2. S IGMOID N ONLINEAR M ULTIVARIATE H AWKES P ROCESSES
Revised: 2. S IGMOID N ONLINEAR M ULTIVARIATE H AWKES P ROCESSES

Original: Similar to the classic nonlinear multivariate Hawkes processes (Br´emaud & Massouli´e, 1996), the i -th dimensional conditional intensity of SNMHP is deﬁned as
Revised: Similar to the classic nonlinear multivariate Hawkes processes (Br´emaud & Massouli´e, 1996), the i -th dimensional conditional intensity of SNMHP is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: where µ i is the base activation of neuron i , h i ( t ) is a real-valued activation and σ ( · ) is the logistic (sigmoid) function which maps the activation into a positive real value in (0 , 1) with λ
Revised: where µ i is the base activation of neuron i , h i ( t ) is a real-valued activation and σ ( · ) is the logistic (sigmoid) function which maps the activation into a positive real value in (0 , 1) with λ

Original: i being a upper-bound to scale it to (0 , λ i ) .
Revised: i being a upper-bound to scale it to (0 , λ i ) .

Original: The sigmoid function is chosen because as seen later, the P´olyaGamma augmentation scheme can be utilized to make the inference tractable.
Revised: The sigmoid function is chosen because as seen later, the P´olyaGamma augmentation scheme can be utilized to make the inference tractable.

Original: After incorporating the nonlinearity, it is straightforward to see the inﬂuence functions, φ ij ( · ) , can be positive or negative.
Revised: After incorporating the nonlinearity, it is straightforward to see the inﬂuence functions, φ ij ( · ) , can be positive or negative.

Original: If φ ij ( · ) is negative, the superposition of φ ij ( · ) will lead to a negative activation
Revised: If φ ij ( · ) is negative, the superposition of φ ij ( · ) will lead to a negative activation

Original: h
Revised: h

Original: i ( t ) that renders the intensity to 0 ; instead, the intensity tends to λ i with a positive φ ij ( · ) .
Revised: i ( t ) that renders the intensity to 0 ; instead, the intensity tends to λ i with a positive φ ij ( · ) .

Original: To achieve a ﬂexible impulse response, the inﬂuence function is assumed to be a weighted sum of basis functions
Revised: To achieve a ﬂexible impulse response, the inﬂuence function is assumed to be a weighted sum of basis functions

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: where { ˜ φ b } Bb =1 are predeﬁned basis functions and w ijb is the weight capturing the inﬂuence from j -th dimension to i -th dimension by b -th basis function with positive indicating excitation and negative indicating inhibition.
Revised: where { ˜ φ b } Bb =1 are predeﬁned basis functions and w ijb is the weight capturing the inﬂuence from j -th dimension to i -th dimension by b -th basis function with positive indicating excitation and negative indicating inhibition.

Original: The basis functions are nonnegative functions capturing the temporal dynamics of the interaction.
Revised: The basis functions are nonnegative functions capturing the temporal dynamics of the interaction.

Original: Although basis functions can be in any form, in order for the weights to represent functional connection strength, basis functions are chosen to be probability densities with compact support that means they have bounded support [0 , T φ ] and the integral is one.
Revised: Although basis functions can be in any form, in order for the weights to represent functional connection strength, basis functions are chosen to be probability densities with compact support that means they have bounded support [0 , T φ ] and the integral is one.

Original: As a result, the i -th dimensional activation is
Revised: As a result, the i -th dimensional activation is

Original: [Equation]
Revised: [Equation]

Original: where Φ jb ( t ) is the convolution of j -th dimensional observation with b -th basis function and can be precomputed; w i
Revised: where Φ jb ( t ) is the convolution of j -th dimensional observation with b -th basis function and can be precomputed; w i

Original: =
Revised: =

Original: [ µ i , w i 11 , . . .
Revised: [ µ i , w i 11 , . . .

Original: , w iMB ] T and Φ ( t ) =
Revised: , w iMB ] T and Φ ( t ) =

Original: [1 , Φ 11 ( t ) , . . .
Revised: [1 , Φ 11 ( t ) , . . .

Original: , Φ MB ( t )] T , both are ( MB + 1) × 1 vectors.
Revised: , Φ MB ( t )] T , both are ( MB + 1) × 1 vectors.

Original: A similar model is used in Linderman (2016) where a binary variable is included to characterize the sparsity of functional connection.
Revised: A similar model is used in Linderman (2016) where a binary variable is included to characterize the sparsity of functional connection.

Original: As shown later, the sparsity in our model is guaranteed by utilizing a Laplace prior on weight instead.
Revised: As shown later, the sparsity in our model is guaranteed by utilizing a Laplace prior on weight instead.

Original: In this paper, the basis functions are scaled (shifted)
Revised: In this paper, the basis functions are scaled (shifted)

Original: Beta densities, but alternatives such as Gaussian or Gamma also can be used.
Revised: Beta densities, but alternatives such as Gaussian or Gamma also can be used.

Original: The reason we choose Beta distribution is the inference of weights will be subject to edge effects with inﬁnite support densities when close to the endpoints of [0 , T φ ] .
Revised: The reason we choose Beta distribution is the inference of weights will be subject to edge effects with inﬁnite support densities when close to the endpoints of [0 , T φ ] .

Original: The weighted sum of Beta densities is a natural choice.
Revised: The weighted sum of Beta densities is a natural choice.

Original: With appropriate mixing, it can be used to approximate functions on bounded intervals arbitrarily well (Kottas, 2006).
Revised: With appropriate mixing, it can be used to approximate functions on bounded intervals arbitrarily well (Kottas, 2006).

Original:  I NFERENCE
Revised:  I NFERENCE

Original: The likelihood of a point process model is provided in Daley & Vere-Jones (2003).
Revised: The likelihood of a point process model is provided in Daley & Vere-Jones (2003).

Original: Correspondingly, the probability density (likelihood) of SNMHP on the i -th dimension as a function of parameters in continuous time is
Revised: Correspondingly, the probability density (likelihood) of SNMHP on the i -th dimension as a function of parameters in continuous time is

Original: [Equation]
Revised: [Equation]

Original: It is worth noting that h i ( t ) depends on w i and observations on all dimensions.
Revised: It is worth noting that h i ( t ) depends on w i and observations on all dimensions.

Original: Our goal is to infer the parameters i.e., weights and intensity upper-bounds, from observations, e.g., neural spike trains, over a time interval [0 , T ] .
Revised: Our goal is to infer the parameters i.e., weights and intensity upper-bounds, from observations, e.g., neural spike trains, over a time interval [0 , T ] .

Original: The functional connectivity in cortical circuits is demonstrated to be sparse in neuroscience (Thomson & Bannister, 2003; Sj¨ostr¨om et al., 2001).
Revised: The functional connectivity in cortical circuits is demonstrated to be sparse in neuroscience (Thomson & Bannister, 2003; Sj¨ostr¨om et al., 2001).

Original: To include sparsity, a factorizing Laplace prior is applied on the weights which characterize the functional connection.
Revised: To include sparsity, a factorizing Laplace prior is applied on the weights which characterize the functional connection.

Original: With the likelihood Eq.
Revised: With the likelihood Eq.

Original: 5 and Laplace prior p L ( w i ) =
Revised: 5 and Laplace prior p L ( w i ) =

Original: (cid:81) j,b
Revised: (cid:81) j,b

Original: 12 α exp ( − | w ijb | α ) , the log-posterior corresponds to a L1 penalized log-likelihood.
Revised: 12 α exp ( − | w ijb | α ) , the log-posterior corresponds to a L1 penalized log-likelihood.

Original: The i -th dimensional MAP estimate can be expressed as
Revised: The i -th dimensional MAP estimate can be expressed as

Original: [Equation]
Revised: [Equation]

Original: where w ∗ i and λ ∗ i are MAP estimates.
Revised: where w ∗ i and λ ∗ i are MAP estimates.

Original: The dependency of the log-posterior on parameters is complex because the sigmoid function exists in the log-likelihood term and the absolute value function exists in the log-prior term.
Revised: The dependency of the log-posterior on parameters is complex because the sigmoid function exists in the log-likelihood term and the absolute value function exists in the log-prior term.

Original: As a result, we have no closed-form solutions for the MAP estimates.
Revised: As a result, we have no closed-form solutions for the MAP estimates.

Original: Numerical optimization methods can be applied, but unfortunately, the efﬁciency is low due to the high dimensionality of parameters which is ( MB + 2) × M .
Revised: Numerical optimization methods can be applied, but unfortunately, the efﬁciency is low due to the high dimensionality of parameters which is ( MB + 2) × M .

Original: To circumvent this issue, three sets of auxiliary latent variables: P´olya-Gamma variables, latent marked Poisson processes and sparsity variables are augmented to make the weights appear in a Gaussian form in the posterior.
Revised: To circumvent this issue, three sets of auxiliary latent variables: P´olya-Gamma variables, latent marked Poisson processes and sparsity variables are augmented to make the weights appear in a Gaussian form in the posterior.

Original: As a result, an efﬁcient EM algorithm with analytical updates is derived to obtain the MAP estimate.
Revised: As a result, an efﬁcient EM algorithm with analytical updates is derived to obtain the MAP estimate.

Original: 3. A UGMENTATION OF P ´ OLYA -G AMMA V ARIABLES
Revised: 3. A UGMENTATION OF P ´ OLYA -G AMMA V ARIABLES

Original: Following Polson et al.
Revised: Following Polson et al.

Original: (2013), the binomial likelihoods parametrized by log odds can be represented as mixtures of Gaussians w.r.t.
Revised: (2013), the binomial likelihoods parametrized by log odds can be represented as mixtures of Gaussians w.r.t.

Original: a P´olya-Gamma distribution.
Revised: a P´olya-Gamma distribution.

Original: Therefore, we can deﬁne a Gaussian representation of the sigmoid function
Revised: Therefore, we can deﬁne a Gaussian representation of the sigmoid function

Original: [Equation]
Revised: [Equation]

Original: (7) where f ( ω, z )
Revised: (7) where f ( ω, z )

Original: =
Revised: =

Original: z/ 2 − z 2 ω/ 2 − log 2 and p PG ( ω | 1 , 0) is the P ´ olya-Gamma distribution with ω ∈ R + .
Revised: z/ 2 − z 2 ω/ 2 − log 2 and p PG ( ω | 1 , 0) is the P ´ olya-Gamma distribution with ω ∈ R + .

Original: Substituting Eq.
Revised: Substituting Eq.

Original: 7 into the likelihood Eq.
Revised: 7 into the likelihood Eq.

Original: 5, the products of observations σ ( h i ( t in )) are transformed into a Gaussian form.
Revised: 5, the products of observations σ ( h i ( t in )) are transformed into a Gaussian form.

Original: 3. A UGMENTATION OF M ARKED P OISSON P ROCESSES
Revised: 3. A UGMENTATION OF M ARKED P OISSON P ROCESSES

Original: Inspired by Donner & Opper (2018), a latent marked Poisson process is augmented to linearize the exponential integral term in the likelihood.
Revised: Inspired by Donner & Opper (2018), a latent marked Poisson process is augmented to linearize the exponential integral term in the likelihood.

Original: Applying the property of sigmoid function σ ( z ) = 1 − σ ( − z ) and Eq.7, the exponential integral term is transformed to
Revised: Applying the property of sigmoid function σ ( z ) = 1 − σ ( − z ) and Eq.7, the exponential integral term is transformed to

Original: [Equation]
Revised: [Equation]

Original: The right hand side is a characteristic functional of a marked Poisson process.
Revised: The right hand side is a characteristic functional of a marked Poisson process.

Original: According to the
Revised: According to the

Original: Campbell’s theorem (Kingman, 2005)
Revised: Campbell’s theorem (Kingman, 2005)

Original: (App. I), the exponential integral term can be rewritten as
Revised: (App. I), the exponential integral term can be rewritten as

Original: [Equation]
Revised: [Equation]

Original: where Π i = { ( ω ik , t ik ) } K i k
Revised: where Π i = { ( ω ik , t ik ) } K i k

Original: =1 denotes a realization of a marked Poisson process and p λ
Revised: =1 denotes a realization of a marked Poisson process and p λ

Original: i is the probability measure of the marked Poisson process Π i with intensity
Revised: i is the probability measure of the marked Poisson process Π i with intensity

Original: λ i ( t, ω )
Revised: λ i ( t, ω )

Original: =
Revised: =

Original: λ i p
Revised: λ i p

Original: PG ( ω
Revised: PG ( ω

Original: | 1 , 0) .
Revised: | 1 , 0) .

Original: The events { t ik } K i k
Revised: The events { t ik } K i k

Original: =1 follow a Poisson process with rate
Revised: =1 follow a Poisson process with rate

Original: λ i and the latent P´olya-Gamma variable ω ik denotes the independent mark at each location t ik .
Revised: λ i and the latent P´olya-Gamma variable ω ik denotes the independent mark at each location t ik .

Original: We can see that, after substituting Eq. 9 into the likelihood
Revised: We can see that, after substituting Eq. 9 into the likelihood

Original: Eq. 5, the exponential integral term is also transformed into a Gaussian form.
Revised: Eq. 5, the exponential integral term is also transformed into a Gaussian form.

Original: 3. A UGMENTATION OF S PARSITY V ARIABLES
Revised: 3. A UGMENTATION OF S PARSITY V ARIABLES

Original: The augmentation of two auxiliary latent variables above makes the augmented likelihood become a Gaussian form w.r.t.
Revised: The augmentation of two auxiliary latent variables above makes the augmented likelihood become a Gaussian form w.r.t.

Original: the weights.
Revised: the weights.

Original: However, the absolute value in the exponent of the Laplace prior hampers the Gaussian form of weights in the posterior.
Revised: However, the absolute value in the exponent of the Laplace prior hampers the Gaussian form of weights in the posterior.

Original: To circumvent this issue, we augment the third set of auxiliary latent variables: sparsity variables.
Revised: To circumvent this issue, we augment the third set of auxiliary latent variables: sparsity variables.

Original: It has been proved that a Laplace distribution can be represented as an inﬁnite mixture of Gaussians (Donner & Opper, 2017; Pontil et al., 2000)
Revised: It has been proved that a Laplace distribution can be represented as an inﬁnite mixture of Gaussians (Donner & Opper, 2017; Pontil et al., 2000)

Original: [Equation]
Revised: [Equation]

Original: where p ( β ijb ) =
Revised: where p ( β ijb ) =

Original: ( β ijb / 2) − 2 exp ( − 1 / (2 β ijb )) .
Revised: ( β ijb / 2) − 2 exp ( − 1 / (2 β ijb )) .

Original: It is straightforward to see the weights are transformed into a Gaussian form in the prior after the augmentation of latent sparsity variables β .
Revised: It is straightforward to see the weights are transformed into a Gaussian form in the prior after the augmentation of latent sparsity variables β .

Original: 3. A UGMENTED L IKELIHOOD AND P RIOR
Revised: 3. A UGMENTED L IKELIHOOD AND P RIOR

Original: After the augmentation of three sets of latent variables, we obtain the augmented joint likelihood and prior (derivation in App. II)
Revised: After the augmentation of three sets of latent variables, we obtain the augmented joint likelihood and prior (derivation in App. II)

Original: [Equation]
Revised: [Equation]

Original: where ω i is the vector of ω in on each t in , β i is a ( MB + 1) × 1 vector of [ β i 00 , β i 11 , . . .
Revised: where ω i is the vector of ω in on each t in , β i is a ( MB + 1) × 1 vector of [ β i 00 , β i 11 , . . .

Original: , β iMB ] T , λ i ( t in , ω in ) =
Revised: , β iMB ] T , λ i ( t in , ω in ) =

Original: λ i p
Revised: λ i p

Original: PG ( ω in | 1 , 0) .
Revised: PG ( ω in | 1 , 0) .

Original: The motivation of augmenting auxiliary latent variables should now be clear: the augmented likelihood and prior contain the weights in a Gaussian form, which corresponds to a quadratic expression for the log-posterior (L1 penalized log-likelihood).
Revised: The motivation of augmenting auxiliary latent variables should now be clear: the augmented likelihood and prior contain the weights in a Gaussian form, which corresponds to a quadratic expression for the log-posterior (L1 penalized log-likelihood).

Original: 3. EM A LGORITHM
Revised: 3. EM A LGORITHM

Original: The original MAP estimate has been represented by Eq. 6.
Revised: The original MAP estimate has been represented by Eq. 6.

Original: With the support of auxiliary latent variables, we propose an analytical EM algorithm to obtain the MAP estimate instead of performing numerical optimization.
Revised: With the support of auxiliary latent variables, we propose an analytical EM algorithm to obtain the MAP estimate instead of performing numerical optimization.

Original: In the standard EM algorithm framework, the lower-bound (surrogate function) of the log-posterior can be represented as
Revised: In the standard EM algorithm framework, the lower-bound (surrogate function) of the log-posterior can be represented as

Original: [Equation]
Revised: [Equation]

Original: with expectation over posterior distributions p (Π i , ω i | w s −
Revised: with expectation over posterior distributions p (Π i , ω i | w s −

Original: [Equation]
Revised: [Equation]

Original: λ s
Revised: λ s

Original: [Equation]
Revised: [Equation]

Original: i ) and p ( β i | w s − 1 i , λ s − 1 i ) , s −indicating parameters from last iteration.
Revised: i ) and p ( β i | w s − 1 i , λ s − 1 i ) , s −indicating parameters from last iteration.

Original: E step : Based on joint distributions in Eq.
Revised: E step : Based on joint distributions in Eq.

Original: 11, the posterior of latent variables can be derived.
Revised: 11, the posterior of latent variables can be derived.

Original: The detailed derivation is provided in App. III.
Revised: The detailed derivation is provided in App. III.

Original: The posterior distributions of P´olya-Gamma variables ω i and sparsity variables β i , and the posterior intensity of marked Poisson process Π i are
Revised: The posterior distributions of P´olya-Gamma variables ω i and sparsity variables β i , and the posterior intensity of marked Poisson process Π i are

Original: [Equation]
Revised: [Equation]

Original: where Λ i ( t, ω ) is the posterior intensity of Π i , p IG is the inverse Gaussian distribution.
Revised: where Λ i ( t, ω ) is the posterior intensity of Π i , p IG is the inverse Gaussian distribution.

Original: It is worth noting that h s − 1 i ( t ) depends on w s − 1 i .
Revised: It is worth noting that h s − 1 i ( t ) depends on w s − 1 i .

Original: The ﬁrst order moments, E [ ω in ] = 1 / (2 h s − 1 i ( t in ))
Revised: The ﬁrst order moments, E [ ω in ] = 1 / (2 h s − 1 i ( t in ))

Original: tanh( h s − 1 i ( t in ) / 2) and E [ β ijb ]
Revised: tanh( h s − 1 i ( t in ) / 2) and E [ β ijb ]

Original: = α/w s − 1 ijb , will be used in the M step.
Revised: = α/w s − 1 ijb , will be used in the M step.

Original: M step : Substituting Eq. 13 into Eq.
Revised: M step : Substituting Eq. 13 into Eq.

Original: 12, we obtain the lower-bound Q ( w i , λ i | w s − 1 i , λ s − 1 i ) .
Revised: 12, we obtain the lower-bound Q ( w i , λ i | w s − 1 i , λ s − 1 i ) .

Original: The updated parameters can be obtained by maximizing the lower-bound.
Revised: The updated parameters can be obtained by maximizing the lower-bound.

Original: The detailed derivation is provided in App. III.
Revised: The detailed derivation is provided in App. III.

Original: Due to the augmentation of auxiliary latent variables, the update of parameters has a closed-form solution
Revised: Due to the augmentation of auxiliary latent variables, the update of parameters has a closed-form solution

Original: [Equation]
Revised: [Equation]

Original: 14a
Revised: 14a

Original: [Equation]
Revised: [Equation]

Original: 14b
Revised: 14b

Original: [Equation]
Revised: [Equation]

Original: where K i =
Revised: where K i =

Original: (cid:82) T 0 (cid:82) ∞ 

Revised: (cid:82) T 0 (cid:82) ∞

Original: Λ i ( t, ω | w s − 1 i , λ s − 1 i ) dωdt ,
Revised: Λ i ( t, ω | w s − 1 i , λ s − 1 i ) dωdt ,

Original: Σ i
Revised: Σ i

Original: =
Revised: =

Original: (cid:104)(cid:82) T 

Revised: (cid:104)(cid:82) T

Original: A i ( t ) Φ ( t ) Φ T ( t ) dt + diag (cid:0)
Revised: A i ( t ) Φ ( t ) Φ T ( t ) dt + diag (cid:0)

Original: α − 2 E
Revised: α − 2 E

Original: [ β i ] (cid:1)(cid:105) −with diag ( · ) indicating the diagonal matrix of a vector, A i ( t )
Revised: [ β i ] (cid:1)(cid:105) −with diag ( · ) indicating the diagonal matrix of a vector, A i ( t )

Original: = (cid:80) N i n =1 E [ ω in ] δ ( t − t in ) + (cid:82) ∞ 0 ω Λ i ( t, ω ) dω , B i ( t ) = 12 (cid:80) N i n =1 δ ( t − t in ) − 12 (cid:82) ∞ 

Revised: = (cid:80) N i n =1 E [ ω in ] δ ( t − t in ) + (cid:82) ∞ 0 ω Λ i ( t, ω ) dω , B i ( t ) = 12 (cid:80) N i n =1 δ ( t − t in ) − 12 (cid:82) ∞

Original: Λ i ( t, ω ) dω with δ ( · ) being the Dirac delta function.
Revised: Λ i ( t, ω ) dω with δ ( · ) being the Dirac delta function.

Original: It is worth noting that numerical quadrature methods, e.g., Gaussian quadrature, need to be applied to intractable integrals above.
Revised: It is worth noting that numerical quadrature methods, e.g., Gaussian quadrature, need to be applied to intractable integrals above.

Original: 3. C OMPLEXITY AND H YPERPARAMETERS
Revised: 3. C OMPLEXITY AND H YPERPARAMETERS

Original: Algorithm
Revised: Algorithm

Original: Result: { λ i ( t )
Revised: Result: { λ i ( t )

Original: =
Revised: =

Original: λ i σ ( w Ti · Φ ( t )) } Mi =
Revised: λ i σ ( w Ti · Φ ( t )) } Mi =

Original: Predeﬁne basis functions { ˜ φ b ( · ) } Bb =1 ;
Revised: Predeﬁne basis functions { ˜ φ b ( · ) } Bb =1 ;

Original: Initialize the hyperparameter α and { λ i , w i , ω i , Π i , β i } Mi =1 ; for Iteration do for Dimension i do
Revised: Initialize the hyperparameter α and { λ i , w i , ω i , Π i , β i } Mi =1 ; for Iteration do for Dimension i do

Original: Update the posterior of ω i by Eq. 13a;
Revised: Update the posterior of ω i by Eq. 13a;

Original: Update the posterior intensity of Π i by Eq. 13b;
Revised: Update the posterior intensity of Π i by Eq. 13b;

Original: Update the posterior of β i by Eq. 13c;
Revised: Update the posterior of β i by Eq. 13c;

Original: Update the intensity upper-bound λ i by Eq. 14a;
Revised: Update the intensity upper-bound λ i by Eq. 14a;

Original: Update the weights w i by Eq. 14b.
Revised: Update the weights w i by Eq. 14b.

Original: end
Revised: end

Original: Update the hyperparameter α .
Revised: Update the hyperparameter α .

Original: end
Revised: end

Original: The complexity of our proposed EM algorithm is O
Revised: The complexity of our proposed EM algorithm is O

Original: ( NN T φ B + L ( N ( MB +
Revised: ( NN T φ B + L ( N ( MB +

Original: [Equation]
Revised: [Equation]

Original: ) 2 + M ( MB + 1) 3 )) where N is the number of observations on all dimensions, N T φ is the the average number of observations on the support of T φ on all dimensions and L is the number of iterations.
Revised: ) 2 + M ( MB + 1) 3 )) where N is the number of observations on all dimensions, N T φ is the the average number of observations on the support of T φ on all dimensions and L is the number of iterations.

Original: The ﬁrst term is due to the convolution nature of Hawkes process, the second and third term to the matrix multiplication and inversion in EM iterations.
Revised: The ﬁrst term is due to the convolution nature of Hawkes process, the second and third term to the matrix multiplication and inversion in EM iterations.

Original: For one application, the number of dimensions M and basis functions B are ﬁxed and much less than N .
Revised: For one application, the number of dimensions M and basis functions B are ﬁxed and much less than N .

Original: Therefore, the complexity can be simpliﬁed as O ( N ( N T φ + L )) .
Revised: Therefore, the complexity can be simpliﬁed as O ( N ( N T φ + L )) .

Original: The hyperparameter α in Laplace prior that encodes the sparsity of weights and parameters of basis functions can be chosen by cross validation or maximizing the lower-bound Q using numerical methods.
Revised: The hyperparameter α in Laplace prior that encodes the sparsity of weights and parameters of basis functions can be chosen by cross validation or maximizing the lower-bound Q using numerical methods.

Original: For the number of basis functions: in essence, a large number leads to a more ﬂexible functional space while a small number results in a faster inference.
Revised: For the number of basis functions: in essence, a large number leads to a more ﬂexible functional space while a small number results in a faster inference.

Original: In experiments, we gradually increase it until no more signiﬁcant improvement.
Revised: In experiments, we gradually increase it until no more signiﬁcant improvement.

Original: Similarly, the number of quadrature nodes and EM iterations is also gradually increased until a suitable value.
Revised: Similarly, the number of quadrature nodes and EM iterations is also gradually increased until a suitable value.

Original: The pseudocode is provided in Alg. 1.
Revised: The pseudocode is provided in Alg. 1.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: We validate the EM algorithm for SNMHP in analyzing both synthetic and real-world spike data collected from the cat primary visual cortex.
Revised: We validate the EM algorithm for SNMHP in analyzing both synthetic and real-world spike data collected from the cat primary visual cortex.

Original: For comparison, the following most relevant baselines are considered: (1) parametric linear multivariate Hawkes processes that are vanilla multivariate
Revised: For comparison, the following most relevant baselines are considered: (1) parametric linear multivariate Hawkes processes that are vanilla multivariate

Original: Hawkes processes with exponential decay inﬂuence functions, for which the inference is performed by MLE (Ozaki, 1979); (2) nonparametric linear multivariate Hawkes processes with ﬂexible inﬂuence functions, for which the inference is by majorization minimization Euler-Lagrange (MMEL)
Revised: Hawkes processes with exponential decay inﬂuence functions, for which the inference is performed by MLE (Ozaki, 1979); (2) nonparametric linear multivariate Hawkes processes with ﬂexible inﬂuence functions, for which the inference is by majorization minimization Euler-Lagrange (MMEL)

Original: (Zhou et al., 2013); (3) parametric nonlinear multivariate Hawkes processes with exponential decay inﬂuence functions, for which the inference is by MCMC based on augmentation and Poisson thinning (MCMC-Aug)
Revised: (Zhou et al., 2013); (3) parametric nonlinear multivariate Hawkes processes with exponential decay inﬂuence functions, for which the inference is by MCMC based on augmentation and Poisson thinning (MCMC-Aug)

Original: (Apostolopoulou et al., 2019).
Revised: (Apostolopoulou et al., 2019).

Original: The implementation of our model is publicly available at https://github.com/zhoufeng6288/SNMHawkesBeta .
Revised: The implementation of our model is publicly available at https://github.com/zhoufeng6288/SNMHawkesBeta .

Original: 4. S YNTHETIC D ATA
Revised: 4. S YNTHETIC D ATA

Original: We analyze spike trains obtained from the synthetic network model shown in Fig.
Revised: We analyze spike trains obtained from the synthetic network model shown in Fig.

Original: 1a.
Revised: 1a.

Original: The synthetic neural network contains four groups of two neurons each.
Revised: The synthetic neural network contains four groups of two neurons each.

Original: In each group, the 2 neurons are selfexciting and mutual-inhibitive while groups are independent of each other.
Revised: In each group, the 2 neurons are selfexciting and mutual-inhibitive while groups are independent of each other.

Original: We assume 4 scaled (shifted)
Revised: We assume 4 scaled (shifted)

Original: Beta distributions as basis functions with support [0 , T φ = 6] in Fig.
Revised: Beta distributions as basis functions with support [0 , T φ = 6] in Fig.

Original: 1b.
Revised: 1b.

Original: For the ground truth, it is assumed that φ 11 = φ 33 = φ 55 = φ 77 = ˜
Revised: For the ground truth, it is assumed that φ 11 = φ 33 = φ 55 = φ 77 = ˜

Original: φ 1 , φ 22 = φ 44 = φ 66 = φ 88 = ˜
Revised: φ 1 , φ 22 = φ 44 = φ 66 = φ 88 = ˜

Original: φ 4 , φ 12 = φ 34 = φ 56 = φ 78 = − 12 ˜ φ 2 , φ 21 = φ 43 = φ 65 = φ 87 = − 12 ˜ φ 3 with positive indicating excitation and negative indicating inhibition.
Revised: φ 4 , φ 12 = φ 34 = φ 56 = φ 78 = − 12 ˜ φ 2 , φ 21 = φ 43 = φ 65 = φ 87 = − 12 ˜ φ 3 with positive indicating excitation and negative indicating inhibition.

Original: With base activation { µ i } 8 i =1 = 0 and upper-bounds { λ i } 8 i =1 = 5 , we use the thinning algorithm (Ogata, 1998) to generate two sets of synthetic spike
Revised: With base activation { µ i } 8 i =1 = 0 and upper-bounds { λ i } 8 i =1 = 5 , we use the thinning algorithm (Ogata, 1998) to generate two sets of synthetic spike

Original: [Table]
Revised: [Table]

Original: !! !" "! "" ## #$
Revised: !! !" "! "" ## #$

Original: [Table]
Revised: [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: 
Revised: 

Original: [Table]
Revised: [Table]

Original: 
Revised: 

Original: [Table]
Revised: [Table]

Original:         1HXURQM         1 HX U RQ  L )XQFWLRQDO&RQQHFWLYLW\(VWLPDWLRQ 
Revised:         1HXURQM         1 HX U RQ  L )XQFWLRQDO&RQQHFWLYLW\(VWLPDWLRQ 

Original:    
Revised:    

Original:  (e) 
Revised:  (e) 

Original: 
Revised: 

Original:    ,WHUDWLRQ í í í   /RJ OL N H OL KRRG H 6\QWKHWLFGDWD/RJOLNHOLKRRG&XUYH WUDLQLQJORJOLNHOLKRRGRI(0WHVWORJOLNHOLKRRGRI( (f)     RITXDGUDWXUHQRGHV     
Revised:    ,WHUDWLRQ í í í   /RJ OL N H OL KRRG H 6\QWKHWLFGDWD/RJOLNHOLKRRG&XUYH WUDLQLQJORJOLNHOLKRRGRI(0WHVWORJOLNHOLKRRGRI( (f)     RITXDGUDWXUHQRGHV     

Original: /RJ OL N H OL KRRG H WUDLQLQJORJOLNHOLKRRGWHVWORJOLNHOLKRRG      5 XQQ L QJ  7 L P H   V  H UXQQLQJWLPH     RIEDVHIXQFWLRQV 
Revised: /RJ OL N H OL KRRG H WUDLQLQJORJOLNHOLKRRGWHVWORJOLNHOLKRRG      5 XQQ L QJ  7 L P H   V  H UXQQLQJWLPH     RIEDVHIXQFWLRQV 

Original:  
Revised:  

Original: /RJ OL N H OL KRRG H WUDLQLQJORJOLNHOLKRRGWHVWORJOLNHOLKRRG     5 XQQ L QJ  7 L P H
Revised: /RJ OL N H OL KRRG H WUDLQLQJORJOLNHOLKRRGWHVWORJOLNHOLKRRG     5 XQQ L QJ  7 L P H

Original:   V  H UXQQLQJWLPH 7UDGHRIIEHWZHHQ$FFXUDF\DQG(IILFLHQF\ (g)        $YHUDJHGRISRLQWVRQHDFKGLPHQVLRQ 
Revised:   V  H UXQQLQJWLPH 7UDGHRIIEHWZHHQ$FFXUDF\DQG(IILFLHQF\ (g)        $YHUDJHGRISRLQWVRQHDFKGLPHQVLRQ 

Original:         6 H F RQG V 5XQQLQJ7LPH (00/(00(/0&0&$XJ (h) data on the time window [0 , T = 1000] with one being the training dataset in Fig. 1c and the other one test dataset in App.
Revised:         6 H F RQG V 5XQQLQJ7LPH (00/(00(/0&0&$XJ (h) data on the time window [0 , T = 1000] with one being the training dataset in Fig. 1c and the other one test dataset in App.

Original: IV.
Revised: IV.

Original: Each dataset contains 8 sequences and each sequence consists of 3340 events on average.
Revised: Each dataset contains 8 sequences and each sequence consists of 3340 events on average.

Original: We aim to identify the functional connectivity of the neural population and the temporal dynamics of inﬂuence functions from statistically dependent spike trains.
Revised: We aim to identify the functional connectivity of the neural population and the temporal dynamics of inﬂuence functions from statistically dependent spike trains.

Original: More experimental details, e.g., hyperparameters, are given in the App. IV.
Revised: More experimental details, e.g., hyperparameters, are given in the App. IV.

Original: The temporal dynamics of interactions among the neural population is shown in Fig.
Revised: The temporal dynamics of interactions among the neural population is shown in Fig.

Original: 1d where we plot the estimated inﬂuence functions of 1-st and 2-nd neurons (other neurons are shown in the
Revised: 1d where we plot the estimated inﬂuence functions of 1-st and 2-nd neurons (other neurons are shown in the

Original: App. IV).
Revised: App. IV).

Original: The estimated ˆ φ 11 and ˆ φ 22 exhibit the self-exciting relation with ˆ φ 12 and ˆ φ 21 characterizing the mutual-inhibitive interactions.
Revised: The estimated ˆ φ 11 and ˆ φ 22 exhibit the self-exciting relation with ˆ φ 12 and ˆ φ 21 characterizing the mutual-inhibitive interactions.

Original: All estimated inﬂuence functions are in a ﬂexible form and close to the ground truth.
Revised: All estimated inﬂuence functions are in a ﬂexible form and close to the ground truth.

Original: Besides, as shown in Fig.
Revised: Besides, as shown in Fig.

Original: 1e, the estimated functional connectivity recovers the ground-truth structure successfully.
Revised: 1e, the estimated functional connectivity recovers the ground-truth structure successfully.

Original: The functional connectivity is deﬁned as (cid:82) | φ ij ( t ) | dt meaning there is no connection only if neither excitation nor inhibition exists.
Revised: The functional connectivity is deﬁned as (cid:82) | φ ij ( t ) | dt meaning there is no connection only if neither excitation nor inhibition exists.

Original: [Table]
Revised: [Table]

Original: The training and test log-likelihood (LogL) curves w.r.t.
Revised: The training and test log-likelihood (LogL) curves w.r.t.

Original: EM iterations are shown in Fig.
Revised: EM iterations are shown in Fig.

Original: 1f where our EM algorithm converges fast with only 50 iterations needed to obtain a plateau.
Revised: 1f where our EM algorithm converges fast with only 50 iterations needed to obtain a plateau.

Original: The trade-off between accuracy (LogL) and efﬁciency (running time) w.r.t.
Revised: The trade-off between accuracy (LogL) and efﬁciency (running time) w.r.t.

Original: the number of quadrature nodes and basis functions is shown in
Revised: the number of quadrature nodes and basis functions is shown in

Original: Fig.
Revised: Fig.

Original: 1g where we can see the accuracy is not sensitive to the number of quadrature nodes over 100 and the optimal number of basis functions is 4.
Revised: 1g where we can see the accuracy is not sensitive to the number of quadrature nodes over 100 and the optimal number of basis functions is 4.

Original: A larger number does not signiﬁcantly improve the accuracy but leads to a longer running time.
Revised: A larger number does not signiﬁcantly improve the accuracy but leads to a longer running time.

Original: Moreover, we compare the running time of our method with alternatives in Fig.
Revised: Moreover, we compare the running time of our method with alternatives in Fig.

Original: 1h where the number of dimensions M is ﬁxed to 2, basis functions B to 4, quadrature nodes to 200 and iterations of all methods to 200.
Revised: 1h where the number of dimensions M is ﬁxed to 2, basis functions B to 4, quadrature nodes to 200 and iterations of all methods to 200.

Original: We can observe that our EM algorithm is the most efﬁcient, even superior to MLE for the classic parametric case, which veriﬁes its efﬁciency.
Revised: We can observe that our EM algorithm is the most efﬁcient, even superior to MLE for the classic parametric case, which veriﬁes its efﬁciency.

Original: Also, we compare our model’s ﬁtting and pre-
Revised: Also, we compare our model’s ﬁtting and pre-

Original: 
Revised: 

Original: 
Revised: 

Original:      T 
Revised:      T 

Original:  1 HX U RQ V 7UDLQLQJ6SLNH7UDLQV (a)    T ϕ (×100 ms ) í í    $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQ ̂ ϕ 8̂    T ϕ (×100 ms ) í í    $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQ ̂ ϕ 8̂    T ϕ (×100 ms ) í í    $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQ ̂ ϕ 9̂    T ϕ (×100 ms ) í í    $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQ ̂ ϕ 9̂ (b) 
Revised:  1 HX U RQ V 7UDLQLQJ6SLNH7UDLQV (a)    T ϕ (×100 ms ) í í    $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQ ̂ ϕ 8̂    T ϕ (×100 ms ) í í    $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQ ̂ ϕ 8̂    T ϕ (×100 ms ) í í    $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQ ̂ ϕ 9̂    T ϕ (×100 ms ) í í    $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQ ̂ ϕ 9̂ (b) 

Original:  1HXURQM  1 HX U RQ  L )XQFWLRQDO&RQQHFWLYLW\(VWLPDWLRQ     
Revised:  1HXURQM  1 HX U RQ  L )XQFWLRQDO&RQQHFWLYLW\(VWLPDWLRQ     

Original:   (c) 
Revised:   (c) 

Original:      ,WHUDWLRQ í í í í í í /RJ OL N H OL KRRG H 5HDOGDWD/RJOLNHOLKRRG&XUYH WUDLQLQJORJOLNHOLKRRGRI(0WHVWORJOLNHOLKRRGRI( (d) diction ability with baseline models for 1-st and 2-nd neurons.
Revised:      ,WHUDWLRQ í í í í í í /RJ OL N H OL KRRG H 5HDOGDWD/RJOLNHOLKRRG&XUYH WUDLQLQJORJOLNHOLKRRGRI(0WHVWORJOLNHOLKRRGRI( (d) diction ability with baseline models for 1-st and 2-nd neurons.

Original: Training and test LogL are shown in Tab. 1 where our SNMHP with EM inference is the champion due to its superior generalized expressive ability.
Revised: Training and test LogL are shown in Tab. 1 where our SNMHP with EM inference is the champion due to its superior generalized expressive ability.

Original: 4. R EAL D ATA
Revised: 4. R EAL D ATA

Original: In this section, we analyze our model performance on a real multi-neuron spike train dataset.
Revised: In this section, we analyze our model performance on a real multi-neuron spike train dataset.

Original: We aim to draw some conclusions about the functional connectivity of cortical circuits and make inferences of the temporal dynamics of inﬂuence.
Revised: We aim to draw some conclusions about the functional connectivity of cortical circuits and make inferences of the temporal dynamics of inﬂuence.

Original: Spike Train Data (Blanche, 2005;
Revised: Spike Train Data (Blanche, 2005;

Original: Apostolopoulou et al., 2019)
Revised: Apostolopoulou et al., 2019)

Original: Several multi-channel silicon electrode arrays are designed to record simultaneously spontaneous neural activity of multiple isolated single units in anesthetized paralyzed cat primary visual cortex areas 17.
Revised: Several multi-channel silicon electrode arrays are designed to record simultaneously spontaneous neural activity of multiple isolated single units in anesthetized paralyzed cat primary visual cortex areas 17.

Original: The spike train dataset contains spike times of 25 simultaneously recorded neurons.
Revised: The spike train dataset contains spike times of 25 simultaneously recorded neurons.

Original: Preliminary Setup We extract the spike times in the time window [0 , 300] (time unit: 100ms, the same applies to the following) as the training data (Fig. 2a) and [300 , 600] as the test data (App. IV).
Revised: Preliminary Setup We extract the spike times in the time window [0 , 300] (time unit: 100ms, the same applies to the following) as the training data (Fig. 2a) and [300 , 600] as the test data (App. IV).

Original: Both datasets contain approximate 7000 timestamps.
Revised: Both datasets contain approximate 7000 timestamps.

Original: All hyperparameters are ﬁne tuned to obtain the maximum test LogL: the scaled (shifted) Beta distribution Beta (˜ α = 50 , ˜ β
Revised: All hyperparameters are ﬁne tuned to obtain the maximum test LogL: the scaled (shifted) Beta distribution Beta (˜ α = 50 , ˜ β

Original: = 50 , shift = − 5) with support [0 , T φ = 10] is designed as the basis function; the number of quadrature nodes is set to 1000 and EM iterations to 100.
Revised: = 50 , shift = − 5) with support [0 , T φ = 10] is designed as the basis function; the number of quadrature nodes is set to 1000 and EM iterations to 100.

Original: More experimental details, e.g., hyperparameters, are given in the
Revised: More experimental details, e.g., hyperparameters, are given in the

Original: App. IV.
Revised: App. IV.

Original: Results 25 × 25 inﬂuence functions among the neuron population are estimated in the application.
Revised: Results 25 × 25 inﬂuence functions among the neuron population are estimated in the application.

Original: An example of the inﬂuence functions between 8-th and 9-th neurons are plotted in Fig.
Revised: An example of the inﬂuence functions between 8-th and 9-th neurons are plotted in Fig.

Original: 2b where our SNMHP model successfully captures the exciting or inhibitive interaction between neurons.
Revised: 2b where our SNMHP model successfully captures the exciting or inhibitive interaction between neurons.

Original: Besides, the estimated functional connectivity is shown in Fig.
Revised: Besides, the estimated functional connectivity is shown in Fig.

Original: 2c where we can see the functional connection structure among neural population is sparse.
Revised: 2c where we can see the functional connection structure among neural population is sparse.

Original: Unfortunately, because the ground-truth functional connectivity of cortical circuits is unknown, the estimated functional connectivity cannot be compared with the ground truth but here we resort to the test LogL to verify whether the estimation is good.
Revised: Unfortunately, because the ground-truth functional connectivity of cortical circuits is unknown, the estimated functional connectivity cannot be compared with the ground truth but here we resort to the test LogL to verify whether the estimation is good.

Original: The training and test LogL curves are shown in Fig.
Revised: The training and test LogL curves are shown in Fig.

Original: 2d where they both reach a close plateau indicating the estimation is appropriate without overﬁtting or underﬁtting.
Revised: 2d where they both reach a close plateau indicating the estimation is appropriate without overﬁtting or underﬁtting.

Original: [Table]
Revised: [Table]

Original: A signiﬁcant advantage of our EM algorithm is the efﬁciency.
Revised: A signiﬁcant advantage of our EM algorithm is the efﬁciency.

Original: The 25-dimensional observation in the real data is a challenge for the inference.
Revised: The 25-dimensional observation in the real data is a challenge for the inference.

Original: For the running time, our EM algorithm costs 3 minutes, the MCMC-Aug costs 1 hour and 45 minutes with the same number of iterations while MLE and
Revised: For the running time, our EM algorithm costs 3 minutes, the MCMC-Aug costs 1 hour and 45 minutes with the same number of iterations while MLE and

Original: MMEL cannot ﬁnish in 2 days due to the curse of dimensionality.
Revised: MMEL cannot ﬁnish in 2 days due to the curse of dimensionality.

Original: Moreover, the ﬁtting and prediction ability is compared in Tab. 2.
Revised: Moreover, the ﬁtting and prediction ability is compared in Tab. 2.

Original: The superior performance of SNMHP w.r.t.
Revised: The superior performance of SNMHP w.r.t.

Original: training and test LogL demonstrates our model can capture the complex mixture of exciting and inhibitive interactions among neural population which leads to better goodness-of-ﬁt.
Revised: training and test LogL demonstrates our model can capture the complex mixture of exciting and inhibitive interactions among neural population which leads to better goodness-of-ﬁt.

Original:  D ISCUSSION AND C ONCLUSION
Revised:  D ISCUSSION AND C ONCLUSION

Original: Although we propose a point-estimation method (EM algorithm) in this work, a straightforward extension to Gibbs sampler is already at hand.
Revised: Although we propose a point-estimation method (EM algorithm) in this work, a straightforward extension to Gibbs sampler is already at hand.

Original: Based on the augmented likelihood and prior, we can obtain the conditional densities of latent variables and parameters in closed form , which constitutes a Gibbs sampler with better efﬁciency than MCMC-Aug since the time-consuming MetropolisHasting sampling in MCMC-Aug is not needed.
Revised: Based on the augmented likelihood and prior, we can obtain the conditional densities of latent variables and parameters in closed form , which constitutes a Gibbs sampler with better efﬁciency than MCMC-Aug since the time-consuming MetropolisHasting sampling in MCMC-Aug is not needed.

Original: However, the proposed Gibbs sampler is less efﬁcient than the proposed EM algorithm because the latent Poisson processes have to be sampled by thinning algorithm in Gibbs sampler which is time consuming.
Revised: However, the proposed Gibbs sampler is less efﬁcient than the proposed EM algorithm because the latent Poisson processes have to be sampled by thinning algorithm in Gibbs sampler which is time consuming.

Original: For the model in Apostolopoulou et al.
Revised: For the model in Apostolopoulou et al.

Original: (2019), a tighter intensity upper-bound is used to reduce the number of thinned points to accelerate the sampler.
Revised: (2019), a tighter intensity upper-bound is used to reduce the number of thinned points to accelerate the sampler.

Original: Instead, our EM algorithm does not encounter this problem as we compute the expectation rather than sampling.
Revised: Instead, our EM algorithm does not encounter this problem as we compute the expectation rather than sampling.

Original: Moreover, Apostolopoulou et al.
Revised: Moreover, Apostolopoulou et al.

Original: (2019) can only use one basis function, which limits inﬂuence functions to be purely exciting or inhibitive exponential decay.
Revised: (2019) can only use one basis function, which limits inﬂuence functions to be purely exciting or inhibitive exponential decay.

Original: Instead, our model can utilize multiple basis functions to characterize an inﬂuence function that is a mixture of excitation and inhibition.
Revised: Instead, our model can utilize multiple basis functions to characterize an inﬂuence function that is a mixture of excitation and inhibition.

Original: In this paper, we develop a SNMHP model in the continuous-time regime which can characterize excitation-inhibition-mixture temporal dependencies among the neural population.
Revised: In this paper, we develop a SNMHP model in the continuous-time regime which can characterize excitation-inhibition-mixture temporal dependencies among the neural population.

Original: Three auxiliary latent variables are augmented to make the corresponding EM algorithm in a closed form to improve efﬁciency.
Revised: Three auxiliary latent variables are augmented to make the corresponding EM algorithm in a closed form to improve efﬁciency.

Original: The synthetic and real data experimental results conﬁrm that our model’s accuracy and efﬁciency are superior to the state of the arts.
Revised: The synthetic and real data experimental results conﬁrm that our model’s accuracy and efﬁciency are superior to the state of the arts.

Original: From the application perspective, although our model is proposed in the neuroscience domain, it can be applied to other applications where the inhibition is a vital factor, e.g., in the coronavirus (COVID-19) spread, the inhibitive effect may represent the medical treatment or cure, or the forced isolation by government.
Revised: From the application perspective, although our model is proposed in the neuroscience domain, it can be applied to other applications where the inhibition is a vital factor, e.g., in the coronavirus (COVID-19) spread, the inhibitive effect may represent the medical treatment or cure, or the forced isolation by government.

Original: From the inference perspective, our EM algorithm is a point-estimation method; other efﬁcient distribution-estimation methods can be developed, e.g., the Gibbs sampler mentioned above or the mean-ﬁeld variational inference.
Revised: From the inference perspective, our EM algorithm is a point-estimation method; other efﬁcient distribution-estimation methods can be developed, e.g., the Gibbs sampler mentioned above or the mean-ﬁeld variational inference.

Original: A CKNOWLEDGMENTS
Revised: A CKNOWLEDGMENTS

Original: The authors would like to thank the anonymous reviewers for insightful comments which greatly improved the paper.
Revised: The authors would like to thank the anonymous reviewers for insightful comments which greatly improved the paper.

Original: This work was partially funded by China Postdoctoral Science Foundation
Revised: This work was supported by NSFC Projects (Nos. 62061136001, 61620106010),

Original: APPENDIX I C AMPBELL ’ S T HEOREM
Revised: Beijing NSF Project (No. JQ19016), Beijing Academy of Artiﬁcial Intelligence (BAAI), TsinghuaHuawei Joint Research Program, a grant from Tsinghua Institute for Guo Qiang, Tiangong Institute for Intelligent Computing and the NVIDIA NVAIL Program with GPU/DGX Acceleration.

Original: Let Π ˆ Z
Revised: F. Zhou was partially funded by China Postdoctoral Science Foundation

Original: =
Revised: APPENDIX I C AMPBELL ’ S T HEOREM

Original: ( z n , ω n ) } Nn =1 be a marked Poisson process on the product space ˆ Z= Z × Ω with intensity Λ( z , ω )
Revised: Let Π ˆ Z

Original: =
Revised: =

Original: Λ( z ) p (  ω | z  ).
Revised: ( z n , ω n ) } Nn =1 be a marked Poisson process on the product space ˆ Z = Z × Ω with intensity Λ( z , ω )

Original: Λ( z ) is the intensity for the unmarked Poisson process { z n } Nn =with ω n ∼ p ( ω n | z n ) being an independent mark drawn at each z n .
Revised: =

Original: Furthermore, we deﬁne a function h ( z , ω  ) :
Revised: Λ( z ) p ( ω | z ) .

Original: Z × Ω → R and the sum H (Π ˆ Z ) 
Revised: Λ( z ) is the intensity for the unmarked Poisson process { z n } Nn = with ω n ∼ p ( ω n | z n ) being an independent mark drawn at each z n .

Original: = (cid:80) ( z , ω ) ∈
Revised: Furthermore, we deﬁne a function h ( z , ω ) :

Original: Π ˆ Zh ( z , ω ).
Revised: Z × Ω → R and the sum H (Π ˆ Z )

Revised: = (cid:80) ( z , ω ) ∈

Revised: Π ˆ Z h ( z , ω ) .

Original: If Λ( z , ω ) < ∞ , then
Revised: If Λ( z , ω ) < ∞ , then

Original: [Equation]
Revised: [Equation]

Original: for any ξ ∈ C .
Revised: for any ξ ∈ C .

Original: The above equation deﬁnes the characteristic functional of a marked Poisson process.
Revised: The above equation deﬁnes the characteristic functional of a marked Poisson process.

Original: This proves Eq.9 in the main paper.
Revised: This proves Eq.9 in the main paper.

Original: The mean is
Revised: The mean is

Original: [Equation]
Revised: [Equation]

Original: which is used when substituting Eq. 13 into Eq.
Revised: which is used when substituting Eq. 13 into Eq.

Original: II D ERIVATION OF A UGMENTED L IKELIHOOD AND P RIOR
Revised: II D ERIVATION OF A UGMENTED L IKELIHOOD AND P RIOR

Original: Substituting Eq.7 and 9 into Eq.5 in the main paper, the augmented likelihood is obtained
Revised: Substituting Eq.7 and 9 into Eq.5 in the main paper, the augmented likelihood is obtained

Original: [Equation]
Revised: [Equation]

Original: where ω i is the vector of ω in and λ i ( t in , ω in )
Revised: where ω i is the vector of ω in and λ i ( t in , ω in )

Original: =
Revised: =

Original: λ i p
Revised: λ i p

Original: PG ( ω in | 1 , 0) .
Revised: PG ( ω in | 1 , 0) .

Original: It is straightforward to see the augmented likelihood is
Revised: It is straightforward to see the augmented likelihood is

Original: [Equation]
Revised: [Equation]

Original: which is Eq.11a.
Revised: which is Eq.11a.

Original: Similarly, the integrand in Eq.
Revised: Similarly, the integrand in Eq.

Original: 10 is just the augmented prior in Eq. 11b.
Revised: 10 is just the augmented prior in Eq. 11b.

Original: III D ERIVATION OF EM A LGORITHM
Revised: III D ERIVATION OF EM A LGORITHM

Original: In the standard EM algorithm framework, the lower-bound of log-posterior has been provided in
Revised: In the standard EM algorithm framework, the lower-bound of log-posterior has been provided in

Original: Eq.
Revised: Eq.

Original: The posterior of latent variables can be derived from the joint distribution in Eq.
Revised: The posterior of latent variables can be derived from the joint distribution in Eq.

Original: The derivation is relatively easy for ω
Revised: The derivation is relatively easy for ω

Original: i and β i while Π i is difﬁcult.
Revised: i and β i while Π i is difﬁcult.

Original: In the following, s − 1 and s mean the last and current iteration in the EM algorithm.
Revised: In the following, s − 1 and s mean the last and current iteration in the EM algorithm.

Original: E S TEP
Revised: E S TEP

Original: The posterior of P´olya-Gamma variables ω
Revised: The posterior of P´olya-Gamma variables ω

Original: i is dependent on the activation h s − 1 i ( t ) at { t in } N i n =
Revised: i is dependent on the activation h s − 1 i ( t ) at { t in } N i n =

Original: , which is further dependent on w s − 1 i through Eq.
Revised: , which is further dependent on w s − 1 i through Eq.

Original: [Equation]
Revised: [Equation]

Original: where we utilize the tilted P´olya-Gamma density p PG ( ω | b, c ) ∝ e − c 2 ω/ 2 p
Revised: where we utilize the tilted P´olya-Gamma density p PG ( ω | b, c ) ∝ e − c 2 ω/ 2 p

Original: PG ( ω | b, 0) (Polson et al., 2013).
Revised: PG ( ω | b, 0) (Polson et al., 2013).

Original: The posterior of sparsity variables β
Revised: The posterior of sparsity variables β

Original: i is an inverse Gaussian distribution which is dependent on weights w s − 

Revised: i is an inverse Gaussian distribution which is dependent on weights w s −

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: The posterior of Π i is dependent on both h s − 

Revised: The posterior of Π i is dependent on both h s −

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: and λ
Revised: and λ

Original: [Equation]
Revised: [Equation]

Original: 1 i
Revised: 1 i

Original: [Equation]
Revised: [Equation]

Original: The Campbell’s theorem can be applied to convert the denominator, the equation above can be transformed as
Revised: The Campbell’s theorem can be applied to convert the denominator, the equation above can be transformed as

Original: [Equation]
Revised: [Equation]

Original: The above posterior distribution is in the likelihood form of a marked Poisson process with intensity function
Revised: The above posterior distribution is in the likelihood form of a marked Poisson process with intensity function

Original: [Equation]
Revised: [Equation]

Original: Substituting posterior distributions of latent variables into Eq.
Revised: Substituting posterior distributions of latent variables into Eq.

Original: 12, we obtain the lower-bound Q .
Revised: 12, we obtain the lower-bound Q .

Original: The ﬁrst term of Eq.
Revised: The ﬁrst term of Eq.

Original: 12 is
Revised: 12 is

Original: [Equation]
Revised: [Equation]

Original: where we utilize the mean rule in Campbell’s theorem
Revised: where we utilize the mean rule in Campbell’s theorem

Original: [Equation]
Revised: [Equation]

Original: C is a constant and
Revised: C is a constant and

Original: [Equation]
Revised: [Equation]

Original: with δ ( · ) being the Dirac delta function and E [ ω in ] = 1 / (2 h s − 1 i ( t in
Revised: with δ ( · ) being the Dirac delta function and E [ ω in ] = 1 / (2 h s − 1 i ( t in

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: 2) (Polson et al., 2013).
Revised: 2) (Polson et al., 2013).

Original: The integral of intensity function has no closed-form solution but can be solved by numerical quadrature methods.
Revised: The integral of intensity function has no closed-form solution but can be solved by numerical quadrature methods.

Original: The second term of Eq.
Revised: The second term of Eq.

Original: 12 is
Revised: 12 is

Original: [Equation]
Revised: [Equation]

Original: where C is a constant, E [ β i ]
Revised: where C is a constant, E [ β i ]

Original: = { E [ β ijb ] } MB +

Revised: = { E [ β ijb ] } MB +

Original: jb
Revised: jb

Original: [Equation]
Revised: [Equation]

Original: and diag ( · ) indicates the diagonal matrix of a vector.
Revised: and diag ( · ) indicates the diagonal matrix of a vector.

Original: The updated parameters λ si and w si can be obtained by setting the gradient of Q to zero. Due to auxiliary variables augmentation, we can see the weights are in a quadratic form in the lower-bound, which leads to an analytical expression
Revised: The updated parameters λ si and w si can be obtained by setting the gradient of Q to zero. Due to auxiliary variables augmentation, we can see the weights are in a quadratic form in the lower-bound, which leads to an analytical expression

Original: [Equation]
Revised: [Equation]

Original: where K
Revised: where K

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: It is worth noting that numerical quadrature methods need to be applied to intractable integrals above.
Revised: It is worth noting that numerical quadrature methods need to be applied to intractable integrals above.

Original: IV E XPERIMENTAL D ETAILS
Revised: IV E XPERIMENTAL D ETAILS

Original: In this appendix, we elaborate on some experimental details.
Revised: In this appendix, we elaborate on some experimental details.

Original: S YNTHETIC D ATA E XPERIMENTS
Revised: S YNTHETIC D ATA E XPERIMENTS

Original: For the synthetic data, the intensities and spike times of our simulated training and test data are shown in Fig.
Revised: For the synthetic data, the intensities and spike times of our simulated training and test data are shown in Fig.

Original: As shown in the experiment of log-likelihood and running time w.r.t.
Revised: As shown in the experiment of log-likelihood and running time w.r.t.

Original: the number of basis functions, the optimal number of basis functions is 4, which are chosen as the ground truth: ˜
Revised: the number of basis functions, the optimal number of basis functions is 4, which are chosen as the ground truth: ˜

Original: φ { 1 , 2 , 3 , 4 } =
Revised: φ { 1 , 2 , 3 , 4 } =

Original: Beta (˜ α
Revised: Beta (˜ α

Original: = 50 , ˜ β
Revised: = 50 , ˜ β

Original: = 50 , scale = 6 , shift = {− 2 , − 1 , 0 , 1 } ) .
Revised: = 50 , scale = 6 , shift = {− 2 , − 1 , 0 , 1 } ) .

Original: By cross validation, the hyperparameter α is chosen to be 0 .
Revised: By cross validation, the hyperparameter α is chosen to be 0 .

Original: As shown in the experiment of log-likelihood and running time w.r.t.
Revised: As shown in the experiment of log-likelihood and running time w.r.t.

Original: the number of quadrature nodes, the accuracy is not sensitive to the number of quadrature nodes over 100, so the number of quadrature nodes is set to 2000.
Revised: the number of quadrature nodes, the accuracy is not sensitive to the number of quadrature nodes over 100, so the number of quadrature nodes is set to 2000.

Original: The number of EM iterations is set to 200 which is large enough for convergence.
Revised: The number of EM iterations is set to 200 which is large enough for convergence.

Original: We plot the estimated inﬂuence functions of 8 neurons in Fig.
Revised: We plot the estimated inﬂuence functions of 8 neurons in Fig.

Original: For comparison, we also plot the estimated inﬂuence functions of 8 neurons from vanilla multivariate Hawkes processes using the MLE algorithm in Fig.
Revised: For comparison, we also plot the estimated inﬂuence functions of 8 neurons from vanilla multivariate Hawkes processes using the MLE algorithm in Fig.

Original: 3 and the functional connectivity graph in Fig.
Revised: 3 and the functional connectivity graph in Fig.

Original: We can see both estimated inﬂuence functions and functional connectivity graph are far from the ground truth.
Revised: We can see both estimated inﬂuence functions and functional connectivity graph are far from the ground truth.

Original: This demonstrates the necessity of incorporating inhibitive interaction into the model when the Hawkes process is applied in the neuroscience domain.
Revised: This demonstrates the necessity of incorporating inhibitive interaction into the model when the Hawkes process is applied in the neuroscience domain.

Original: The running time experiment and the ﬁtting and prediction experiment are both conducted for 2 neurons because the baseline models cannot ﬁnish in 2 days with 8 neurons because of the curse of dimensionality.
Revised: The running time experiment and the ﬁtting and prediction experiment are both conducted for 2 neurons because the baseline models cannot ﬁnish in 2 days with 8 neurons because of the curse of dimensionality.

Original:         1 HX U RQ   ,QWHQVLWLHVDQGVSLNHVRIQHXURQV         1 HX U RQ           1 HX U RQ           1 HX U RQ   LQWHQVLW\ VSLNHV         1 HX U
Revised:         1 HX U RQ   ,QWHQVLWLHVDQGVSLNHVRIQHXURQV         1 HX U RQ           1 HX U RQ           1 HX U RQ   LQWHQVLW\ VSLNHV         1 HX U

Original: RQ           1 HX U RQ           1 HX U RQ         7UDLQLQJ'DWDVHW   1 HX U RQ           1 HX U RQ   ,QWHQVLWLHVDQGVSLNHVRIQHXURQV         1 HX U RQ           1 HX U RQ           1 HX U RQ   LQWHQVLW\ VSLNHV         1 HX U
Revised: RQ           1 HX U RQ           1 HX U RQ         7UDLQLQJ'DWDVHW   1 HX U RQ           1 HX U RQ   ,QWHQVLWLHVDQGVSLNHVRIQHXURQV         1 HX U RQ           1 HX U RQ           1 HX U RQ   LQWHQVLW\ VSLNHV         1 HX U

Original: RQ          
Revised: RQ          

Original: [Figure]
Revised: [Figure]

Original: U
Revised: U

Original: [Figure]
Revised: [Figure]

Original:       
Revised:       

Original: [Figure]
Revised: [Figure]

Original: 
Revised: 

Original: [Figure]
Revised: [Figure]

Original: U RQ     
Revised: U RQ     

Original: [Figure]
Revised: [Figure]

Original:        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 11 1214 1517 18 11 *7 12 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 21 2224 2527 28 21 *7 22 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 31 3234 3537 38 33 *7 34 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 41 4244 4547 48 43 *7 44 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 51 5254 5557 58 55 *7 56 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 61 6264 6567 68 65 *7 66 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 71 7274 7577 78 77 *7 78 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 81 8284 8587 88 87 *7 88 *
Revised:        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 11 1214 1517 18 11 *7 12 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 21 2224 2527 28 21 *7 22 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 31 3234 3537 38 33 *7 34 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 41 4244 4547 48 43 *7 44 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 51 5254 5557 58 55 *7 56 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 61 6264 6567 68 65 *7 66 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 71 7274 7577 78 77 *7 78 *        T      $ F W L Y D W L RQ ,QIOXHQFHIXQFWLRQVRI1HXURQ 81 8284 8587 88 87 *7 88 *

Original: 6 T 0.
Revised: 6 T 0.

Original: A c t i v a t i o n
Revised: A c t i v a t i o n

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: 6 T 0.
Revised: 6 T 0.

Original: A c t i v a t i o n
Revised: A c t i v a t i o n

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: n
Revised: n

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: 6 T 0.
Revised: 6 T 0.

Original: A c t i v a t i o n
Revised: A c t i v a t i o n

Original: Influence functions of Neuron77 78 77 GT 78 GT      6 T 0. A c t i v a t i o n
Revised: Influence functions of Neuron77 78 77 GT 78 GT      6 T 0. A c t i v a t i o n

Original: Influence functions of Neuron87 88 87 GT 88 GT
Revised: Influence functions of Neuron87 88 87 GT 88 GT

Original: R EAL D ATA E XPERIMENTS
Revised: R EAL D ATA E XPERIMENTS

Original: For the real spike data in cat primary visual cortex areas 17, it contains spike times of 25 simultaneously recorded neurons.
Revised: For the real spike data in cat primary visual cortex areas 17, it contains spike times of 25 simultaneously recorded neurons.

Original: We extract the spike times in the time window [0 , 300] (time unit: 100ms) as the training data and [300 , 600] as the test data. Both datasets contain approximate 7000 timestamps.
Revised: We extract the spike times in the time window [0 , 300] (time unit: 100ms) as the training data and [300 , 600] as the test data. Both datasets contain approximate 7000 timestamps.

Original: The training and test spike trains are plotted in Fig.
Revised: The training and test spike trains are plotted in Fig.

Original: 5 below. All hyperparameters are ﬁne tuned in real data experiments.
Revised: 5 below. All hyperparameters are ﬁne tuned in real data experiments.

Original: Speciﬁcally, the optimal basis function is chosen as: ˜ φ =
Revised: Speciﬁcally, the optimal basis function is chosen as: ˜ φ =

Original: Beta (˜ α
Revised: Beta (˜ α

Original: = 50 , ˜ β
Revised: = 50 , ˜ β

Original: = 50 , scale = 10 , shift = − 5) .
Revised: = 50 , scale = 10 , shift = − 5) .

Original: The hyperparameter α is optimised to be 0 .
Revised: The hyperparameter α is optimised to be 0 .

Original: 1 by cross validation.
Revised: 1 by cross validation.

Original: The number of quadrature nodes is chosen to be 1000 for which the running time is acceptable.
Revised: The number of quadrature nodes is chosen to be 1000 for which the running time is acceptable.

Original: The number of EM iterations is set to 100 which is large enough for convergence.
Revised: The number of EM iterations is set to 100 which is large enough for convergence.

Original: [Figure]
Revised: [Figure]

Original: n
Revised: n

Original: [Figure]
Revised: [Figure]

Original: Functional Connectivity: MLE Estimation
Revised: Functional Connectivity: MLE Estimation

Original: [Figure]
Revised: [Figure]

Original: 
Revised: 

Original: 
Revised: 

Original:      T  
Revised:      T  

Original: [Figure]
Revised: [Figure]

Original: 
Revised: 

Original:     
Revised:     

Original: [Figure]
Revised: [Figure]

Original:  
Revised:  

Original: [Figure]
Revised: [Figure]