text:
\section{Introduction}
\label{sec:INT}

Recently, much attention has been paid to brain rhythms, observed in electrical recordings of firing activity \cite{Buz}. These brain rhythms emerge via synchronization between firings of individual neurons. This kind of neural synchronization may be used for efficient sensory and cognitive processing \cite{Wang1,Wang2}, and it is also correlated with pathological rhythms associated with neural diseases \cite{Disease1,Disease2,Disease3}. Here, we are interested in characterization of population synchronization of bursting neurons in terms of neural synchrony measures \cite{Order}. Bursting occurs when neuronal activity alternates, on a slow timescale, between a silent phase and an active (bursting) phase of fast repetitive spikings \cite{Rinzel1,Rinzel2,Burst1,Burst2,Burst3}. Due to the slow and fast timescales of bursting activity, bursting neurons exhibit two types of burst and spike synchronizations. Burst synchronization on the slow bursting timescale refers to a coherence between the active phase onset or offset times of bursting neurons, while spike synchronization on the fast spike timescale characterizes a coherence between intraburst spikes fired by bursting neurons  \cite{Burstsync1,Burstsync2}. Many recent studies on the burst and spike synchronizations have been made in several aspects (e.g., chaotic phase synchronization, transitions between different states of burst synchronization, effect of network topology, effect on information transmission, suppression of bursting synchronization, and effect of noise and coupling on the burst and spike synchronization) \cite{BSsync1,BSsync2,BSsync3,BSsync4,BSsync5,BSsync6,BSsync7,BSsync8,BSsync9,BSsync10,BSsync11,BSsync12,BSsync13,BSsync14,BSsync15}.

In this paper, we are interested in practical characterization of the burst and spike synchronization transitions of bursting neurons in the frequency domain. Population synchronization may be well visualized in the raster plot of neural spikes which can be obtained in experiments. Instantaneous population firing rate (IPFR), $R(t)$, which is directly obtained from the raster plot of spikes, is a realistic population quantity describing collective behaviors in both the computational and the experimental neuroscience \cite{Wang1,Sparse1,Sparse2,Sparse3,Sparse4,Sparse5,Sparse6}. This experimentally-obtainable $R(t)$ is in contrast to the ensemble-averaged potential $X_G$ which is often used as a population quantity in the computational neuroscience, because to directly get $X_G$ in real experiments is very difficult. To overcome this difficulty, instead of $X_G$, we used $R(t)$ as a population quantity, and developed a realistic order parameter, based on $R(t)$, to make practical characterization of synchronization of spiking neurons in both the computational and the experimental neuroscience \cite{Kim}. The mean square deviation of $R(t)$ plays the role of the realistic order parameter $\cal {O}$ used to determine the threshold value for the synchronization transition of spiking neurons. In this way, synchronization transition of spiking neurons may be well characterized in terms of the realistic order parameter $\cal {O}$, based on the IPFR $R(t)$.

In the field of neuroscience, power-spectral analysis of a time-series $x(t)$ (e.g., neuronal membrane potential) is often made to examine how the variance of the data $x(t)$ is distributed over the frequency components into which $x(t)$ may be decomposed \cite{Sparse1,Sparse2,Sparse3,Sparse4,Sparse5,Sparse6}. Following this conventional direction, we investigate the synchronization transitions of bursting neurons in the frequency domain. The main purpose of our works is to characterize the synchronization transitions of bursting neurons in terms of ``frequency-domain'' order parameters by extending the realistic ``time-domain'' order parameter of spiking neurons \cite{Kim} to the case of bursting neurons. This extension work on the frequency-domain order parameters is in contrast to another extension work where the synchronization transitions of bursting neurons are characterized in terms of the time-domain order parameters \cite{Kim1}. The IPFR $R(t)$ shows the whole combined population behaviors with both the slow and fast timescales. To clearly investigate the synchronization transitions of bursting neurons, we separate the slow and fast timescales of the bursting activity via frequency filtering, and decompose the IPFR $R(t)$ into $R_b(t)$ (the instantaneous population burst rate (IPBR) describing the bursting behavior) and $R_s(t)$ (the instantaneous population spike rate (IPSR) describing the intraburst spiking behavior). In presence of the burst and spike synchronizations, $R_b$ and $R_s$ exhibit regular oscillations, independently of $N$ (the number of the bursting neurons). On the other hand, in absence of the burst and spike synchronizations, $R_b$ and $R_s$ become stationary as $N$ goes to the infinity. The synchronous oscillations of $R_b$ and $R_s$ in the time domain may be well characterized by the bursting and spiking peaks of their power spectral densities. As in the case of the coherence resonance \cite{CR1}, each spectral ``resonance'' (i.e., peak) in the power spectral density may be well analyzed in terms of a ``coherence factor'' $\beta$ (i.e., a measure of spectral coherence) which is defined by a ``signal to noise'' ratio of the spectral peak height and its relative width \cite{CR2,CR3}. We also note that the signal to noise ratio has a long history of being used in neuroscience as a measure of the fidelity of signal transmission and detection by neurons and synapses \cite{CR4}. Then, the bursting and spiking coherence factors $\beta_b$ and $\beta_s$ of the bursting and spiking peaks in the power spectral densities of $R_b(t)$ and $R_s(t)$ are shown to play the role of the bursting and spiking order parameters in the frequency domain which are used to determine the bursting and spiking thresholds for the bursting and spiking transitions, respectively. We also consider another raster plot of bursting onset or offset times which visualizes the bursting behaviors more directly. From this type of raster plot, we may directly obtain the IPBR, $R_b^{(on)}(t)$ or $R_b^{(off)}(t)$, without frequency filtering. Then, the bursting onset and offset coherence factors, $\beta_b^{(on)}$ and $\beta_b^{(off)}$, of the bursting onset and offset peaks in the power spectral densities of $R_b^{(on)}(t)$ and $R_b^{(off)}(t)$ are also shown to play the role of the frequency-domain bursting order parameters for the bursting transition. The frequency-domain order parameters $ \beta_b^{(on)}$ and $\beta_b^{(off)}$ yield the same bursting threshold which is obtained through calculation of $\beta_b$, and they are more direct ones than $\beta_b$ because they may be directly obtained without frequency filtering. Consequently, all the frequency-domain bursting and spiking order parameters may be usefully used for characterization of the burst and and spike synchronization transitions of the bursting neurons in the frequency domain.

This paper is organized as follows. In Sec.~\ref{sec:HR}, as an example for characterization we describe an inhibitory network of bursting Hindmarsh-Rose (HR) neurons \cite{HR1,HR2,HR3,HR6}. In Sec.~\ref{sec:OR}, we separate the slow bursting and the fast spiking timescales via frequency filtering, and develop realistic frequency-domain bursting and spiking order parameters (i.e., the bursting and spiking coherence factors), based on the power spectral densities of the IPBR and the IPSR, which are applicable in both the computational and the experimental neuroscience.  Their usefulness for characterization of the burst and spike synchronization transitions is shown in explicit examples of bursting HR neurons. Finally, a summary is given in Section \ref{sec:SUM}.

\section{A Network of Inhibitory Bursting Hindmarsh-Rose Neurons}
\label{sec:HR}
As an example for characterization, we consider an inhibitory network of $N$ globally-coupled bursting HR neurons. The representative bursting HR neuron model was originally introduced to describe the time evolution of the membrane potential for the pond snails \cite{HR1,HR2,HR3,HR6}. The population dynamics in this inhibitory network is governed by the following set of ordinary differential equations:
\begin{eqnarray}
\frac{dx_i}{dt} &=& y_i - a x^{3}_{i} + b x^{2}_{i} - z_i +I_{DC} +D \xi_{i} -I_{syn,i}, \label{eq:CHRA} \\
\frac{dy_i}{dt} &=& c - d x^{2}_{i} - y_i, \label{eq:CHRB} \\
\frac{dz_i}{dt} &=& r \left[ s (x_i - x_o) - z_i \right], \label{eq:CHRC} \\
\frac{dg_i}{dt}&=& \alpha g_{\infty}(x_i) (1-g_i) - \beta g_i, \;\;\; i=1, \cdots, N, \label{eq:CHRD}
\end{eqnarray}
where
\begin{eqnarray}
I_{syn,i} &=& \frac{J}{N-1} \sum_{j(\ne i)}^N g_j(t) (x_i - X_{syn}), \label{eq:CHRE} \\
g_{\infty} (x_i) &=& 1/[1+e^{-(x_i-x_s^*)\delta}]. \label{eq:CHRF}
\end{eqnarray}
Here, the state of the $i$th HR neuron at a time $t$ (measured in units of milliseconds) is described by four state variables: the fast membrane potential $x_i$, the fast recovery current $y_i,$ the slow adaptation current $z_i$, and the synaptic gate variable $g_i$ representing the fraction of open synaptic ion channels. The parameters in the single HR neuron are taken as $a=1.0,$ $b=3.0,$ $c=1.0,$ $d=5.0,$ $r=0.001,$ $s=4.0,$  and $x_o=-1.6$ \cite{CR3}.

Each bursting HR neuron is stimulated by using the common DC current $I_{DC}$ and an independent Gaussian white noise $\xi_i$ [see the 5th and the 6th terms in Eq.~(\ref{eq:CHRA})] satisfying $\langle \xi_i(t) \rangle =0$ and $\langle \xi_i(t)~\xi_j(t') \rangle = \delta_{ij}~\delta(t-t')$, where
$\langle\cdots\rangle$ denotes the ensemble average. The noise $\xi$ is a parametric one that randomly perturbs the strength of the applied current
$I_{DC}$, and its intensity is controlled by using the parameter $D$. As $I_{DC}$ passes a threshold $I_{DC}^* (\simeq 1.26)$ in the absence of noise, each single HR neuron exhibits a transition from a resting state to a bursting state. Throughout this paper, we consider the suprathreshold case of $I_{DC}=1.3$
where each HR neuron exhibits spontaneous bursting activity without noise. Figures \ref{fig:Single}(a)-\ref{fig:Single}(b) show the time series of the fast membrane potential $x(t)$ and the fast recovery current $y(t)$, while the time series of the slow adaptation current $z(t)$ is shown in Fig.~\ref{fig:Single}(c). As seen well in the time series of $x$ and $y$, the bursting activity alternates, on a slow timescale, between a silent phase and an active (bursting) phase of fast repetitive spikings. For this case, the slow bursting timescale is $\tau_b \simeq 609$ ms [corresponding to the slow bursting frequency $f_b$ $(\simeq 1.6$ Hz)], while the fast spiking timescale is $\tau_s \simeq 18.2$ ms [corresponding to the fast spiking frequency $f_s$ $(\simeq 55$ Hz)].

The last term in Eq.~(\ref{eq:CHRA}) represents the synaptic coupling of the network. $I_{syn,i}$ of Eq.~(\ref{eq:CHRE}) represents a synaptic current injected into the $i$th neuron. Here the coupling strength is controlled by the parameter $J$ and $X_{syn}$ is the synaptic reversal potential. Here, we use $X_{syn}=-2$ for the inhibitory synapse. The synaptic gate variable $g$ obeys the 1st order kinetics of Eq.~(\ref{eq:CHRD}) \cite{KI1,KI2}. Here, the normalized concentration of synaptic transmitters, activating the synapse, is assumed to be an instantaneous sigmoidal function of the membrane potential with a threshold $x_s^*$ in Eq.~(\ref{eq:CHRF}), where we set $x_s^*=0$ and $\delta=30$ \cite{HR5}.
The transmitter release occurs only when the neuron emits a spike (i.e., its potential $x$ is larger than $x^*_s$). For the inhibitory GABAergic synapse (involving the $\rm{GABA_A}$ receptors), the synaptic channel opening rate, corresponding to the inverse of the synaptic rise time $\tau_r$, is $\alpha=10$ ${\rm ms}^{-1}$, and the synaptic closing rate $\beta$, which is the inverse of the synaptic decay time $\tau_d$, is $\beta=0.1$ ${\rm ms}^{-1}$ \citep{GABA1,GABA2}. Hence, $I_{syn}$ rises fast and decays slowly.

Numerical integration of Eqs.~(\ref{eq:CHRA})-(\ref{eq:CHRD}) is done using the Heun method \cite{SDE} (with the time step $\Delta t=0.01$ ms).
For each realization of the stochastic process, we choose a random initial point $[x_i(0),y_i(0),z_i(0),g_i(0)]$ for the $i$th $(i=1,\dots, N)$ neuron with uniform probability in the range of $x_i(0) \in (-2,2)$, $y_i(0) \in (-16,0)$, $z_i(0) \in (1.1,1.4)$, and $g_i(0) \in (0,1)$.

\section{Frequency-domain order parameters for the burst and spike synchronization transitions}
\label{sec:OR}
In this section, we extend the realistic order parameter of spiking neurons to the case of bursting neurons for characterization of population synchronization transition in the frequency domain. For our aim, we separate the slow bursting and the fast spiking timescales through frequency filtering, and decompose the IPFR $R(t)$ into the IPBR $R_b(t)$ (describing  the bursting behavior) and the IPSR $R_s(t)$ (describing the intraburst spiking behavior). Then, we develop realistic frequency-domain bursting and spiking order parameters, based on the power spectral densities of the IPBR $R_b(t)$ and the IPSR $R_s(t)$, and show their usefulness for characterization of the burst and spike synchronization transitions in explicit examples of bursting HR neurons.

As an example for characterization, we consider an inhibitory network of $N$ globally-coupled bursting HR neurons, and characterize the synchronization transitions of bursting HR neurons
in the frequency domain by varying the noise intensity $D$. To compare our results in the frequency domain with those in the time domain, we fix the DC current strength $I_{DC}$ and the coupling strength $J$ at $I_{DC}=1.3$ and $J=0.3$, as in the time-domain work \cite{Kim1}. In computational neuroscience, a population-averaged global potential,
\begin{equation}
 X_G (t) = \frac {1} {N} \sum_{i=1}^{N} x_i(t),
\label{eq:GPOT}
\end{equation}
is often used for describing emergence of population synchronization. In this study, we consider the population behaviors after the transient time of $2 \times 10^3$ ms. Although the global potential $X_G$ is an important ensemble-averaged quantity to describe synchronization in computational neuroscience, it is practically difficult to directly get $X_G$ in real experiments. To overcome this difficulty, instead of $X_G$, we use the IPFR which is an experimentally-obtainable population quantity used in both the experimental and the computational neuroscience \cite{Wang1,Sparse1,Sparse2,Sparse3,Sparse4,Sparse5,Sparse6}. The IPFR is obtained from the raster plot of spikes which is a collection of spike trains of individual neurons. Such raster plots of spikes, where population synchronization may be well visualized, are fundamental data in the experimental neuroscience. The raster plots of spikes in Figs.~\ref{fig:Bursting1}(a1)-\ref{fig:Bursting1}(a5) show population states for various values of noise intensity $D$. To get a smooth IPFR from the raster plot of spikes, we employ the kernel density estimation (kernel smoother) \cite{Kernel}. Each spike in the raster plot is convoluted (or blurred) with a kernel function $K_h(t)$ to get a smooth estimate of IPFR, $R(t)$:
\begin{equation}
R(t) = \frac{1}{N} \sum_{i=1}^{N} \sum_{s=1}^{n_i} K_h (t-t_{s}^{(i)}),
\label{eq:IPSRK}
\end{equation}
where $t_{s}^{(i)}$ is the $s$th spiking time of the $i$th neuron, $n_i$ is the total number of spikes for the $i$th neuron, and we use a Gaussian
kernel function of band width $h$:
\begin{equation}
K_h (t) = \frac{1}{\sqrt{2\pi}h} e^{-t^2 / 2h^2}, ~~~~ -\infty < t < \infty.
\label{eq:Gaussian}
\end{equation}
Figures \ref{fig:Bursting1}(b1)-\ref{fig:Bursting1}(b5) show smooth IPFR kernel estimates $R(t)$ of band width $h=1$ ms for $D=0$, 0.01, 0.04, 0.06 and 0.08, respectively. For $D=0$, clear ``bursting bands,'' each of which is composed of ``stripes'' of spikes, appear successively at nearly regular time intervals [see Fig.~\ref{fig:Bursting1}(a1)]; a magnified 1st intraburst band is given in Fig.~\ref{fig:Spiking}(a1). For the case of $D=0$, both the burst synchronization [synchrony on the slow bursting timescale $\tau_b$ ($\simeq 215$ ms)] and the spike synchronization [synchrony on the fast spike timescale $\tau_s$ $(\simeq 14.6$ ms)] occur in each bursting band. As a result of this complete synchronization, the IPFR kernel estimate $R(t)$ shows a bursting activity [i.e., fast spikings appear on a slow wave in $R(t)$], as shown in Fig.~\ref{fig:Bursting1}(b1). However, as $D$ is increased, loss of spike synchronization occurs in each bursting band because spiking stripes become smeared due to a destructive role of noise. As an example, see the case of $D=0.01$ where the raster plot of spikes and the IPFR kernel estimate $R(t)$ are shown in Figs.~\ref{fig:Bursting1}(a2) and \ref{fig:Bursting1}(b2), respectively. The magnified 1st bursting band in Fig.~\ref{fig:Spiking}(a3) shows smearing of the spiking stripes well. Consequently, the amplitude of $R(t)$ decreases, as shown in Fig.~\ref{fig:Bursting1}(b2). As $D$ is further increased and passes a spiking noise threshold $D^*_s$ $(\simeq 0.032)$, complete loss of spike synchronization occurs in each bursting band. Then, only the burst synchronization (without spike synchronization) occurs, as shown in the case of $D=0.04$ in Figs.~\ref{fig:Bursting1}(a3) and \ref{fig:Bursting1}(b3). For this case, $R(t)$ shows a slow-wave oscillation without spikes. With increase in $D$, such ``incoherent'' bursting bands become more and more smeared, and hence the degree of burst synchronization decreases [e.g., see the case of $D=0.06$ in Fig.~\ref{fig:Bursting1}(a4)]. As a result, the amplitude of $R(t)$ is further decreased, as shown in Fig.~\ref{fig:Bursting1}(b4) for $D=0.06$. With further increasing $D$, incoherent bursting bands begin to overlap, which eventually results in the complete loss of burst synchronization as $D$ passes another larger bursting noise threshold $D^*_b$ $(\simeq 0.068)$. Consequently, for $D> D^*_b$, completely unsynchronized states with nearly stationary $R(t)$ appear, as shown in the case of $D=0.08$ in Figs.~\ref{fig:Bursting1}(a5) and \ref{fig:Bursting1}(b5).

The (above) IPFR kernel estimate $R(t)$ is a population quantity describing the ``whole'' combined collective behaviors of bursting neurons with both the slow bursting and the fast spiking timescales. Through frequency filtering, we separate the slow and the fast timescales, and decompose the IPFR kernel estimate $R(t)$ into the IPBR $R_b(t)$ and the IPSR $R_s(t)$ for more clear investigation of the burst and spike synchronizations. Through band-pass filtering of $R(t)$ [with the lower and the higher cut-off frequencies of 3 Hz (high-pass filter) and 7 Hz (low-pass filer)], we get the regularly-oscillating IPBR $R_b(t)$ (containing only the slow wave without spikes) in Figs.~\ref{fig:Bursting1}(c1)-\ref{fig:Bursting1}(c5) for $D=0$, 0.01, 0.04, 0.06, and 0.08. As $D$ is increased, the amplitude of $R_b(t)$ decreases gradually, and eventually $R_b(t)$ becomes nearly stationary when $D$ passes the bursting noise threshold $D^*_b$ $(\simeq 0.068)$. We note that synchronous oscillations of $R_b(t)$ in the time domain are characterized by the bursting peaks in the power spectral densities of $\Delta R_b(t)$ $[=R_b(t)- \overline{R_b(t)}]$, where the overline represents the time average. Figures \ref{fig:Bursting1}(d1)-\ref{fig:Bursting1}(d5) show distinct bursting peaks in the power spectra of $\Delta R_b(t)$; each power spectrum is made of $2^{15}$ data points and smoothed through the Daniell filters of length 3 and 5 \cite{PS}. Then, each bursting peak may be analyzed well in terms of a bursting coherence factor $\beta_b$ defined by the product of the height $H_p$ and the $Q$ factor of the peak \cite{CR1,CR2,CR3}:
\begin{equation}
\beta_b = H_p~Q; Q = f_p / \Delta f_p.
\end{equation}
Here, $f_p$ and $\Delta f_p$ are the frequency of the bursting peak and the width of the bursting peak at the height of $e^{-1/2}~h$, respectively. For more accurate results, we repeat the process to get the bursting coherence factor $\beta_b$ for multiple realizations. Thus, we obtain ${\langle \beta_b \rangle}_r$ (average bursting coherence factor) through average over 20 realizations. Figure \ref{fig:Bursting1}(e) shows plots of the average bursting coherence factor ${\langle \beta_b \rangle}_r$ versus $D$. For $D < D^*_b$ $(\simeq 0.068$), synchronized bursting states exist because the values of ${\langle \beta_b \rangle}_r$ become saturated to non-zero limit values in the thermodynamic limit of $N \rightarrow \infty$ (i.e., bursting peaks persist, independently of $N$). However, as $D$ passes the bursting noise threshold $D^*_b$, the average bursting coherence factor ${\langle \beta_b \rangle}_r$ tends to zero as $N \rightarrow \infty$ (i.e., eventually bursting peaks disappear in the thermodynamic limit), and hence a transition to unsynchronized bursting states occurs because the noise spoils the burst synchronization completely. In this way, the average bursting coherence factor ${\langle \beta_b \rangle}_r$ describes the burst synchronization transition well in the frequency domain, and hence it plays the role of the realistic frequency-domain bursting order parameter for the bursting transition (i.e., one can determine the bursting noise threshold $D^*_b$ through calculation of ${\langle \beta_b \rangle}_r$). This frequency-domain bursting order parameter ${\langle \beta_b \rangle}_r$ is in contrast to the time-domain bursting order parameter, based on the time-averaged fluctuation of the IPBR $R_b(t)$ \cite{Kim1}. In spite of their difference, calculations of both the frequency-domain and the time-domain bursting order parameters result in the same bursting noise threshold $D^*_b$ (compare Fig.~\ref{fig:Bursting1}(e) with Fig.~3(a) in \cite{Kim1}). Consequently, the frequency-domain bursting order parameter may be used effectively to determine $D^*_b$ for the bursting transition, like the case of the time-domain bursting order parameter.

From now on, we investigate the intraburst spike synchronization transition of bursting HR neurons in the frequency domain by varying the noise intensity $D$.
Figures \ref{fig:Spiking}(a1)-\ref{fig:Spiking}(a5) and Figures \ref{fig:Spiking}(b1)-\ref{fig:Spiking}(b5) show the raster plots of intraburst spikes and the corresponding (band-pass filtered) IPSR $R_s(t)$ during the 1st global bursting cycle of the IPBR $R_b(t)$, respectively for various values of $D$: synchronized spiking states for $D=0$, 0.005, 0.01, and 0.02, and unsynchronized spiking state for $D=0.06$. Here, the IPSRs $R_s(t)$ are obtained through band-pass filtering of the IPFR kernel estimate $R(t)$ [with the lower and the higher cut-off frequencies of 30 Hz (high-pass filter) and 90 Hz (low-pass filer)]. Then, the intraburst spike synchronization may be well described in terms of $R_s(t)$. For $D=0$, clear 8 spiking stripes (composed of spikes and indicating population spike synchronization) appear in the intraburst band of the 1st global bursting cycle of $R_b(t)$ in Fig.~\ref{fig:Spiking}(a1), and the band-pass filtered IPSR $R_s(t)$ shows only the fast spiking oscillations (without a slow wave) with the population spiking frequency $f_s$ $(\simeq 68.5$ Hz) in Fig.~\ref{fig:Spiking}(b1). However, as $D$ is increased, spiking stripes in the intraburst band become more and more smeared (e.g., see the cases of $D=0.005$, 0.01, and 0.02). Consequently, the amplitude of $R_s(t)$ decreases due to loss of spike synchronization. Eventually, when $D$ passes the spiking noise threshold $D^*_s$ $(\simeq 0.032)$, spikes become completely scattered within the intraburst band, and $R_s(t)$ becomes nearly stationary. Consequently, for $D> D^*_s$, complete loss of spike synchronization occurs in the intraburst band, as shown in Fig.~\ref{fig:Spiking}(b5) for $D=0.06$.
Figures \ref{fig:Spiking}(c1)-\ref{fig:Spiking}(c5) show the power spectra of $\Delta R_s(t)$ $[=R_s(t)- \overline{R_s(t)}]$ in the 1st global bursting cycle of $R_b(t)$: each power spectrum is made of $2^{8}$ data points and smoothed through the Daniell filters of length 3 and 5. Spiking peaks in their power spectra are analyzed in terms of the spiking coherence factors $\beta_s$ (defined by the product of the height $H_p$ and the $Q$ factor of the peak). For more accurate results, we repeat the process to get $\beta_s$ for multiple realizations. In each realization we follow the 20 global bursting cycles of $R_b(t)$, and get the double-averaged spiking coherence factor ${\langle {\langle \beta_s \rangle}_b \rangle}_r$ through average over 20 realizations. Figure \ref{fig:Spiking}(d) shows plots of the double-averaged spiking coherence factor ${\langle {\langle \beta_s \rangle}_b \rangle}_r$ versus $D$. For $D < D^*_s$ $(\simeq 0.032$), synchronized spiking states exist because the values of ${\langle {\langle \beta_s \rangle}_b \rangle}_r$ become saturated to non-zero limit values as $N \rightarrow \infty$ (i.e., spiking peaks persist, irrespectively of $N$). However, when $D$ passes the spiking noise threshold $D^*_s$, ${\langle {\langle \beta_s \rangle}_b \rangle}_r$ tends to zero in the thermodynamic limit of $N \rightarrow \infty$ (i.e., eventually spiking peaks disappear in the thermodynamic limit), and hence a transition to unsynchronized spiking states occurs because the noise spoils the intraburst spike synchronization completely. In this way, the double-averaged spiking coherence factor ${\langle {\langle \beta_s \rangle}_b \rangle}_r$ describes the intraburst spike synchronization transition well in the frequency domain, and hence it plays the role of the realistic frequency-domain spiking order parameter for the spiking transition (i.e., one can determine the spiking noise threshold $D^*_s$ through calculation of ${\langle {\langle \beta_s \rangle}_b \rangle}_r$). This frequency-domain spiking order parameter is also in contrast to the time-domain spiking order parameter, based on the time-averaged fluctuation of the IPSR $R_s(t)$ \cite{Kim1}. We also note that both the frequency-domain and  the time-domain spiking order parameters yield the same spiking noise threshold $D^*_s$ (compare Fig.~\ref{fig:Spiking}(d) with Fig.~6(d) in \cite{Kim1}). Consequently, the frequency-domain spiking order parameter may also be used effectively to determine $D^*_s$ for the spiking transition, as in the case of the time-domain spiking order parameter.

Finally, we consider another raster plot of bursting onset or offset times for more direct visualization of bursting behavior.
[At the onset (offset) times of the $i$th bursting HR neuron, its individual potential $x_i$ passes the threshold of $x^*_b = -1$ from below (above).] Without frequency filtering, we can directly obtain the IPBR kernel estimate, $R_b^{(on)}(t)$ [$R_b^{(off)}(t)$] from the raster plot of the bursting onset (offset) times. Figures \ref{fig:Bursting2}(a1)-\ref{fig:Bursting2}(a5) show the raster plots of the bursting onset times for various values of $D$, while the raster plots of the bursting offset times are shown in Figs.~\ref{fig:Bursting2}(c1)-\ref{fig:Bursting2}(c5). From these raster plots of the bursting onset (offset) times, we obtain smooth IPBR kernel estimates, $R_b^{(on)}(t)$ [$R_b^{(off)}(t)$], of band width $h=50$ ms in Figs.~\ref{fig:Bursting2}(b1)[(d1)]-\ref{fig:Bursting2}(b5)[(d5)] for $D=0$, 0.01, 0.04, 0.06, and 0.08. For $D=0$, clear bursting ``stripes'' [composed of bursting onset (offset) times and indicating burst synchronization] appear successively at nearly regular time intervals; the bursting onset and offset stripes are time-shifted [see Figs.~\ref{fig:Bursting2}(a1) and \ref{fig:Bursting2}(c1)]. The corresponding IPBR kernel estimates, $R_b^{(on)}(t)$ and $R_b^{(off)}(t)$, for $D=0$ show regular oscillations with the same population bursting frequency $f_b$ $(\simeq 4.7$ Hz), although they are phase-shifted [see Figs.~\ref{fig:Bursting2}(b1) and \ref{fig:Bursting2}(d1)]. With increasing $D$, the bursting onset and offset stripes in the raster plots become smeared and begin to overlap, and thus the degree of the burst synchronization decreases. As a result, the amplitudes of both $R_b^{(on)}(t)$ and $R_b^{(off)}(t)$ decrease gradually (e.g., see the cases of $D=0.01$, 0.04, and 0.06). Eventually, as $D$ passes the bursting noise threshold $D^*_b$ $(\simeq 0.068)$, bursting onset and offset times become completely scattered in the raster plots, and the corresponding IPBR kernel estimates, $R_b^{(on)}(t)$ and $R_b^{(off)}(t)$, become nearly stationary, as shown in Figs.~\ref{fig:Bursting2}(b5) and \ref{fig:Bursting2}(d5) for $D=0.08$. Figures \ref{fig:Bursting2}(e1)-\ref{fig:Bursting2}(e5) show the power spectra of $\Delta R_b^{(on)}(t)$ $[=R_b^{(on)}(t)- \overline{R_b^{(on)}(t)}]$, while Figures \ref{fig:Bursting2}(f1)-\ref{fig:Bursting2}(f5) show the power spectra of $\Delta R_b^{(off)}(t)$ $[=R_b^{(off)}(t)- \overline{R_b^{(off)}(t)}]$; each power spectrum is made of $2^{15}$ data points and smoothed through the Daniell filters of length 3 and 5. Bursting onset and offset peaks in these power spectra are analyzed in terms of the bursting onset and offset coherence factors $\beta_b^{(on)}$ and $\beta_b^{(off)}$ (each coherence factor of a peak is defined by the product of the height $h$ and the $Q$ factor of the peak). For more accurate results, we repeat the process to obtain $\beta_b^{(on)}$ and $\beta_b^{(off)}$ for multiple realizations. Thus, we obtain ${\langle \beta_b^{(on)} \rangle}_r$ (average bursting onset coherence factor) and ${\langle \beta_b^{(off)} \rangle}_r$ (average bursting offset coherence factor) through average over 20 realizations. Figures \ref{fig:Bursting2}(g1) and \ref{fig:Bursting2}(g2) show plots of the average bursting onset and offset coherence factor ${\langle \beta_b^{(on)} \rangle}_r$ and ${\langle \beta_b^{(off)} \rangle}_r$ versus $D$, respectively. As in the case of the (above) average bursting coherence factor ${\langle \beta_b \rangle}_r$, when passing the same bursting noise threshold $D^*_b$ $(\simeq 0.068)$, both the average bursting onset and offset coherence factors ${\langle \beta_b^{(on)} \rangle}_r$ and ${\langle \beta_b^{(off)} \rangle}_r$ go to zero as $N \rightarrow \infty$ (i.e., eventually bursting onset and offset peaks disappear in the thermodynamic limit), and hence a transition to burst unsynchronization occurs for $D > D^*_b$, because the noise breaks up the burst synchronization completely. In this way, both the average bursting onset and offset coherence factors, ${\langle \beta_b^{(on)} \rangle}_r$ and ${\langle \beta_b^{(off)} \rangle}_r$, describe the burst synchronization transition well in the frequency domain, and hence they also play the role of the realistic frequency-domain bursting order parameters for the bursting transition together with ${\langle \beta_b \rangle}_r$. These frequency-domain bursting order parameters are also in contrast to the time-domain bursting order parameters, based on the time-averaged fluctuations of
$R_b^{(on)}(t)$ and $R_b^{(off)}(t)$ \cite{Kim1}. We note that both the frequency-domain and the time-domain bursting order parameters yield the same bursting noise threshold $D^*_b$ (compare Figs.~\ref{fig:Bursting2}(g1) and \ref{fig:Bursting2}(g2) with Figs.~3(b) and 3(c) in \cite{Kim1}). Consequently, along with
${\langle \beta_b \rangle}_r$, the frequency-domain bursting order parameters, ${\langle \beta_b^{(on)} \rangle}_r$ and ${\langle \beta_b^{(off)} \rangle}_r$, may also be used effectively to determine $D^*_b$ for the bursting transition, as in the case of the time-domain bursting order parameters.

\section{Summary} \label{sec:SUM}
We have extended the realistic time-domain order parameter of spiking neurons to the case of bursting neurons. Their usefulness for characterization of the burst and spike synchronization transitions in the frequency domain has been shown in explicit examples of bursting HR neurons by varying the noise intensity $D$. Population synchronization may be well visualized in the raster plot of neural spikes which may be obtained in experiments. The IPFR kernel estimate $R(t)$, which is obtained from the raster plot of spikes, is a realistic collective quantity describing the whole combined population behaviors with the slow bursting and the fast spiking timescales. Through frequency filtering, we have decomposed the IPFR kernel estimate $R(t)$ into the IPBR $R_b(t)$ and the IPSR $R_s(t)$. We note that both $R_b(t)$ and $R_s(t)$ may be used to effectively characterize the burst and spike synchronizations, respectively. For synchronous cases, oscillations of $R_b$ and $R_s$ in the time domain are characterized by the bursting and spiking peaks in their power spectral densities. Similar to the case of coherence resonance, each spectral resonance (i.e., peak) may be well analyzed in terms of the coherence factor $\beta$, defined by a ``signal to noise'' ratio of the spectral peak height and its relative width. The average bursting and spiking coherence factors ${\langle \beta_b \rangle}_r$ and ${\langle {\langle \beta_s \rangle}_b \rangle}_r$ of the bursting and spiking peaks in the power spectral densities of $\Delta R_b$ and $\Delta R_s$ have been found to play the role of the frequency-domain bursting and spiking order parameters for the burst and spike synchronization transitions, respectively. Through calculation of ${\langle \beta_b \rangle}_r$ and ${\langle {\langle \beta_s \rangle}_b \rangle}_r$, we have determined the noise bursting and spiking thresholds, $D^*_b$ and $D^*_s$, beyond which the burst and spike synchronizations break up, respectively. For more direct visualization of bursting behavior, we consider another raster plot of bursting onset or offset times, from which the IPBR, $R_b^{(on)}(t)$ or $R_b^{(off)}(t)$, can be directly obtained without frequency filtering. Then, the average bursting onset and offset coherence factors, ${\langle \beta_b^{(on)} \rangle}_r$ and ${\langle \beta_b^{(off)} \rangle}_r$ of the bursting onset and offset peaks in the power spectral densities of $\Delta R_b^{(on)}(t)$ and $\Delta R_b^{(off)}(t)$ have also been shown to play the role of the frequency-domain bursting order parameters for the bursting transition. These frequency-domain order parameters ${\langle \beta_b^{(on)} \rangle}_r$ and ${\langle \beta_b^{(off)} \rangle}_r$ yield the same bursting noise threshold $D^*_b$ which is obtained via calculation of ${\langle \beta_b \rangle}_r$, and they are more direct ones than ${\langle \beta_b \rangle}_r$ because they may be directly obtained without frequency filtering. We also note that all these bursting and spiking noise thresholds are the same as those obtained through calculations of the time-domain bursting and spiking order parameters \cite{Kim1}. Consequently, the frequency-domain bursting and spiking order parameters may be usefully used for characterizing the burst and spike synchronization transitions of the bursting neurons, as in the case of the time-domain bursting and spiking order parameters.

\begin{acknowledgments}
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2013057789).
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


LONDON (Reuters) - Unilever said on Friday it had agreed to buy Procter & Gamble's Fluocaril and Parogencyl toothpaste brands for an undisclosed price, expanding its presence in oral care.
The brands are sold primarily in France and Spain.
Unilever expects the deal to close in the second quarter.


The Federal Acquisition Regulation (FAR) allows several opportunities for the government to provide feedback to bidders during or after competitions. The post-award debriefing of offerors is one of those opportunities, and can be a very valuable tool for companies seeking feedback on their proposals.
The government is required by the FAR to provide a post-award debriefing to any offeror who requests one in writing within 3 days of notification of contract award.
In the dozens of competition debriefs I've conducted or attended in more than a decade, I'm consistently surprised by how often we repeat the same information. The following reviews the format we use for debriefings, questions we're frequently asked during the discussions, and some of the common feedback we seem to repeat regularly.
The Department of Defense (DoD) guidance on debriefings states the objective as: "The crux of any post award debriefing is the SSA [Source Selection Authority] award decision and whether that decision is well supported and resulted from a source selection conducted in a thorough, fair and sound manner consistent with the requirements and source-selection methodology established in the RFP [request for proposal]." The preceding quote (under section B.8.3.1) and other information about DoD source selections can be found on DAU's Acquisition Community Connection at: https://acc.dau.mil/dodssp.
First, our standard debriefing format: The objective of this post-award debriefing is to highlight the significant elements in your proposal and to summarize the rationale for award. The ground rules are open and honest discussions within the limits of FAR 15.506.
The focus is on your proposal submission. But overall evaluated cost, task order management and technical proposal ranking for the successful bidder will be provided, including summary of the rationale for award.
Reasonable responses will be given to relevant questions about whether the source-selection procedures, applicable regulations and other applicable authorities were followed in eliminating your proposal from the competition.
A color-code rating technique was used to evaluate the Management and Technical proposals. Past Performance was evaluated for an overall confidence rating and cost proposals were not given a rating. Each proposal was evaluated against the following four factors: (1) Management, (2) Technical Proposal, (3) Past Performance, and (4) Cost. Evaluation of Factors 1 and 2 focused on the strengths, weaknesses, significant weaknesses and deficiencies of the proposals. Evaluation of risk associated with the proposals for these factors are inherent in the evaluation.
As outlined within the RFP, Management and Technical are equal in importance and more important than Past Performance. When combined, these three are significantly more important than Cost.
Following the reading of our standard debriefing, we review the ratings the company in question received. In particular, we focus on the "strengths, weaknesses, significant weaknesses, and deficiencies of the proposal" that resulted in the final overall rating.
Q: Can you tell us how we might compete more favorably next time?
A: Our response to this generally is fairly standard, and tracks directly back to what we tell you in Sections L (Instructions, conditions, and notices to offerors or respondents) and M (Evaluation factors for award). First, your proposal should show that you understand the requirement, preferably without regurgitating it. Second, your proposal should demonstrate how you are going to meet the requirement. Last, but certainly not least, the higher color ratings are awarded when the proposal (1) meets requirements; (2) shows a thorough (or exceptional) approach and understanding of the requirements; (3) contains strengths which outweigh (or far outweigh) any weaknesses; (4) and when risk—not evaluated separately—of unsuccessful performance is low or very low.
Q: Why wasn't our "concept X" evaluated as a strength?
A: The DoD source-selection procedures (https://acc.dau. mil/dodssp) define a strength as "an aspect of an offeror's proposal that has merit or exceeds specified performance or capability requirements in a way that will be advantageous to the government during contract performance." It is incumbent on the vendors to demonstrate their understanding of the requirement, and explain how their approaches will provide value to the government. In many cases, good ideas do not rise to the level of a strength in evaluation because: (1) the concept expressed in the proposal does provide value to the government but is part of what was asked for in the RFP (i.e., is part of how you will meet our requirements, not a way to meet them better, smarter, faster, etc.); or (2) the concept isn't supported by or integrated with the rest of the proposal (does not track to pricing, is not supported by staffing, is not integrated with service-delivery model, etc.). For example, nearly all proposals we review include ideas such as reach-back support, a council of graybeards to provide strategic consultation, or something else intended to differentiate the proposal from others. But, without providing details on the specific, tangible outcomes (in terms of hours, work products or deliverables) that meet the definition of strength, the government will not evaluate them as strengths during a source selection.
A: In general, we would prefer that it never come to this. Our intent is to have significant and substantive discussions throughout our acquisitions to the broadest extent authorized. As a result of those discussions, we should at the very least have communicated to the vendors any significant deficiencies or weaknesses in their proposals and given them time to correct those deficiencies. The presence of a weakness in the final evaluation generally means (1) we don't believe the vendor understands or recognizes the weakness we've pointed out and hasn't changed its proposal to respond to it; or (2) despite the vendor's attempt(s) to respond to the weakness, we still don't understand how the vendor plans to address it or don't see the staffing or other resources to resolve the matter.
Q: Wasn't this just a Lowest Priced, Technically Acceptable (LPTA) source selection?
A: There is a time and place for LPTA, but the RFP will always state specifically where the evaluation falls on the best value continuum. The vast majority of our source selections are conducted as best value trade-offs. From the top down in Special Operations Research, Development and Acquisition, we're strong believers in best value source selections and actively strive to be the best in DoD at conducting them. We focus a great deal of time and effort to ensure we have a well-trained and prepared acquisition workforce with the experience and tools to properly execute, document and communicate the source selections we make and to defend the selections in the event of any protests.
Q: Can you tell us how our cost or proposal compared with the other offerors?
A: Unfortunately, no. In most cases, we will provide the winning offeror's total cost, and the winner's evaluation results in terms of colors. We are prohibited by the FAR from disclosing any proprietary information (including other offerors' costs), directly comparing vendors or providing point-bypoint comparisons.
The evaluation team felt you spent too much of your proposal regurgitating the requirement to us. It's sometimes a fine balance, but you need to convey to us that you understand the requirement without just reading it back to us. In addition, including examples of work on past efforts does not demonstrate your understanding of the requirement. That experience is evaluated as part of past performance.
Your pricing, staffing model or overall approach (or portions of them) did not make sense to us, were not well supported or didn't track back clearly to your understanding of the requirement. When evaluating your proposal, we take a very structured approach. We read to understand your overall approach and understanding of the requirement, evaluate whether your proposal meets our requirements, and then identify any strengths or weaknesses of your approach. Well-written proposals lead us clearly and unambiguously through that process and are consistent throughout. An example of this is dividing a large proposal into sections by different vendor offices or organizations. This can save time by having the subject-matter expert write each proposal area, but frequently results in a disjointed proposal when the different sections are not well integrated. We recommend a detailed final review by the offeror of the entire proposal to ensure it is clear and consistent and that the data are not repeated in multiple sections.
Evaluation of past performance is based on the offeror's recent/relevant performance record from a variety of sources. This may include information provided by the offeror, information obtained from questionnaires (internally or externally), or information obtained from any other source available to the government (Past Performance Information Retrieval System, electronic Subcontract Reporting System, etc.).
So, that's a quick down and dirty overview of the format we use for debriefings of unsuccessful offerors, questions we're frequently asked during the discussions, and some of the common feedback we seem to repeat regularly. Hopefully, it provides some insight into the thought patterns and work processes of the evaluation team and background for your next source selection.
The author can be contacted at anthony.davis@socom.mil.
Now, thanks to Georgia Tech's workshop, a GSA proposal can be actually completed at the workshop. If a business is not prepared to submit all the documentation at the time of the workshop, the GSA proposal preparation process easily can be shortened to within 30 days following the workshop.
By attending the workshop, business people will save time and money with an instructor-guided, do-It-yourself approach as well as avoid mistakes that typically delay or stop GSA Schedule proposals from being considered. Attendees will receive expert guidance and valuable instructional documents, including templates and sample narratives based on successful GSA Schedule offers. Fifteen (15) Continuing Professional Education (CPE) credits are also granted to each person successfully completing the workshop.
The GSA Schedule Proposal Preparation Workshop is conducted by experienced government contracting professionals who have helped many businesses successfully land GSA Schedule contracts. Real world business scenarios are presented during this instructional session, tackling all the difficult and confusing aspects of the proposal process.
As an added bonus, The Academy provides workshop attendees with four hours of time with instructional personnel – after the workshop – to answer any remaining questions, review proposal packages, and provide follow-on consulting.
The St. Louis GSA Schedule Proposal Preparation Workshop will be held on the Florissant Valley campus of St. Louis Community College, located in north St. Louis County. The workshop is being hosted by the Missouri Procurement Technical Assistance Centers.
To register for this one-of-a-kind Workshop, please click here or visit: http://tinyurl.com/kv2by74.
The reviews are in! Participants in the GSA Schedule Proposal Preparation Workshop unanimously agree that the workshop exceeded their expectations.
On January 13 and 14, 2014, The Contracting Education Academy at Georgia Tech (The Academy) hosted its first workshop devoted to helping businesses put together a GSA Schedule proposal package. The hands-on instruction enables participants to actually complete their proposals during the two-day workshop.
The next opportunity to attend this unique workshop is May 19 and 20, 2014. The workshop is scheduled to be conducted in Atlanta, Georgia on Georgia Tech's midtown campus. Registration details for the next workshop can be found here.
Real world business scenarios are presented during this instructional workshop, tackling all the difficult and confusing aspects of the proposal process. The Academy's affordable course is designed to reduce the typical proposal preparation time frame. Our workshop is conducted by experienced government contracting professionals who have helped many business people successfully land GSA Schedule contracts.
The Academy's GSA Schedule Proposal Preparation Workshop provides your business with the ability to complete your Schedule in as little as two days, depending on how readily available your proposal documentation is. Even if you are just starting your proposal, our GSA schedule workshop can shorten the preparation process to as little as 30 days.
Plus, as an added bonus, The Academy provides workshop attendees with four hours of time with our instructional personnel – after the workshop – to answer any remaining questions, review your proposal package, and provide follow-on consulting.
More details, including the course outline, can be found here. Registration details are here.


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3 Biotech Stocks We Love
By Sean Williams , Todd Campbell , and Cheryl Swanson Fool.com
Source: Flickr user Frankieleon.
One of the oldest Wall Street idioms is that you should "never fall in love with a stock." While we adhere to this idiom a good deal of the time, sometimes we just can't help but shoot Cupid's arrow in the direction of a company we believe has the tools to outperform for a very long time.
With that in mind, we asked three of our top healthcare analysts to share one biotech stock they've fallen head over heels for -- metaphorically speaking, of course. Here's what they had to say.
I can't help myself, I've fallen forCelgene and perhaps you might fall for it, too. That's because Celgene isn't just a market-share-leading maker of oncology medicine, it's also one of the most shareholder-friendly biotechnology companies out there, returning 347% since 2010.
Source: Celgene.
Celgene's success stems from Revlimid, a leading second-line therapy for multiple myeloma that's on pace to eclipse $5.6 billion in sales this year. Celgene also markets cancer drug Abraxane and third-line multiple myeloma drug Pomalyst, as well as fast-growing psoriasis drug Otezla.
Across all of these medicines, Celgene expects to deliver sales and profits of at least $9 billion and $4.75 per share this year, respectively. If so, then there should be plenty of cash kicking around to keep buying back shares, investing in R&D, and gobbling up smaller companies, such as itsrecent purchaseof promising MS drug developer Receptos.
Celgene's top tier product lineup has led to a bulletproof balance sheet, but what makes Celgene even more attractive to me is its long-term forecast.
In January, Celgene offered up guidance for sales of at least $20 billion by 2020, and last month, Celgene upped that estimate to $21 billion. Celgene also increased its 2020 earnings forecast from at least $12.50 per share to $13 per share. Although a lot can go wrong between now and then, a forecast like that suggests I'm justified in wanting a long-term relationship with Celgene's shares.
Gilead Sciences is a stock I've changed my mind about recently. Like my spouse of 19 years, I've come to realize Gilead is no flash in the pan -- it's a keeper.
Source: Gilead Sciences.
Backlash from payers and competition from rival drugs have made this stock volatile. Yet -- despite all the fuss -- Gilead now hasa 90% market sharein hepatitis C patients. Meanwhile, the stock trades at 10.4 times forward earnings, making it one of the cheapest biotechs you can buy.
Last quarter, Gilead not only crushed analyst expectations, it raised revenue and lowered expense guidance. You can't ask for better than that. The outlook was raised from $29 billion to $30 billion for full-year 2015. The lower end of its full-year profit margin forecast was raised to 88% from 87%, while the top end was left unchanged at 90%.
But what makes Gilead truly extraordinary is how it has revolutionized whole segments of healthcare. And it's done so at an amazing pace thanks to scarily smartacquisitions. The recent acquisitions of EpiTherapeutics and Phenex Pharmaceuticals are likely just priming the pump. With a cash hoard of over $14 billion and a knack for knowing what company owns the next world-beating drug, Gilead is a stock to keep from now until your golden years. What's not to love?
Sean WilliamsAs for me, I'm going to show some love to the midcap biotech space and point Cupid's arrow atMedivation.
The mantle piece for Medivation is cancer drug Xtandi, which was developed in collaboration withAstellas Pharma. After being approved by the Food and Drug Administration nearly three months ahead of its PDUFA decision date in 2012 as a treatment for previously treated patients with metastatic castration-resistant prostate cancer, Xtandi achieved a potentially more lucrative label expansion last year when it was given the green light by the FDA as a first-line indication prior to chemotherapy.
Source: Astellas Pharma.
The PREVAIL trial demonstrated that Xtandi reduced the risk of death by 29% and radiographic progression by 83%, while delaying the median time mCRPC patients had before beginning chemotherapy by a whopping 17 months, or close to triple the time compared to the placebo.
This statistically significant improvement in the second most-commonly diagnosed cancer type has meant big bucks for Medivation and its partner Astellas. In the first quarter, Medivation announced that total Xtandi sales grew 161% year over year in international markets, and rose 6% in the U.S., largely due to a 9% reduction from negative currency translation.
Looking ahead, Xtandi has the potential to expand its label to treat triple-negative breast cancer, and it could double its sales from $710 million in 2014 to more than $1.6 billion by 2018, all while its EPS is projected to grow from $3.42 in 2014 to $7.42 by 2018.
Though competition may remain fierce in mCRPC, Xtandi looks like it'll remain a star for years to come, and that could mean big gains for Medivation as demand for Xtandi grows.
The article 3 Biotech Stocks We Love originally appeared on Fool.com.
Cheryl SwansonandSean Williams have no position in any stocks mentioned in this article. Todd Campbell owns shares of Celgene and Gilead Sciences. The Motley Fool owns shares of and recommends Celgene and Gilead Sciences. Try any of our Foolish newsletter services free for 30 days. We Fools may not all hold the same opinions, but we all believe that considering a diverse range of insights makes us better investors. The Motley Fool has a disclosure policy.
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