Death ligand concentration and the membrane proximal signaling module regulate the type 1/type 2 choice in apoptotic death signaling Abstract Apoptotic death pathways are frequently activated by death ligand induction and subsequent activation of the membrane proximal signaling module. Death receptors cluster upon binding to death ligands, leading to formation of a membrane proximal death-inducing-signaling-complex (DISC). In this membrane proximal signalosome, initiator caspases (caspase 8) are processed resulting in activation of both type 1 and type 2 pathways of apoptosis signaling. How the type 1/type 2 choice is made is an important question in the systems biology of apoptosis signaling. In this study, we utilize a Monte Carlo based in silico approach to elucidate the role of membrane proximal signaling module in the type 1/type 2 choice of apoptosis signaling. Our results provide crucial mechanistic insights into the formation of DISC signalosome and caspase 8 activation. Increased concentration of death ligands was shown to correlate with increased type 1 activation. We also study the caspase 6 mediated system level feedback activation of apoptosis signaling and its role in the type 1/type 2 choice. Our results clarify the basis of cell-to-cell stochastic variability in apoptosis activation and ramifications of this issue is further discussed in the context of therapies for cancer and neurodegenerative disorders. Apoptosis, a programmed mode of cell death, is used in a variety of cellular and physiological situations ranging from developmental regulations to tissue homeostasis. Apoptosis is frequently mediated by death ligands (DL) that are known to cluster death receptors (DR) and membrane proximal adaptor proteins in a signaling complex called death-inducing-signaling-complex (DISC). Procaspase 8 is recruited by adaptor molecules in the DISC and gets cleaved to its active form caspase 8, which in turn, activates both type 1 and type 2 pathways of apoptosis. However, a clear mechanistic understanding of DISC formation and caspase 8 activation, under death ligand induction, is lacking. In addition to its relevance in the biology of apoptosis, such mechanistic understanding can be key to developing therapies for apoptosis related diseases. Death ligand (such as FasL) induced apoptosis has been shown to be the mode of cell death in activation induced cell death (ADCC) in lymphocytes (Elmore 2007) . In addition, killing of pathogen infected cells or tumor cells by cytotoxic T lymphocytes (CTLs) and natural killer (NK) cells is frequently mediated by death ligands (Elmore 2007) . TRAIL, a death ligand found on the plasma membrane of NK cells (Takeda et al. 2001) , has emerged as a promising agent in cancer chemotherapy (Falschlehner et al. 2007; Shirley et al. 2011; Kurita et al. 2011; Picarda et al. 2012) . Recent experiments have shown that the fraction of melanoma (skin cancer) stem cells is much higher in NOD SCID IL2rg-/-mice than that found in regular NOD SCID xenotransplantation assays (Quintana et al. 2008) . Mice lacking the IL2rg receptor chain are known to be depleted of NK cells, thus, indicating an important role for TRAIL mediated killing of tumor cells. DISC formation and Caspase 8 activation have also been implicated in neurodegenerative disorders such as in Alzheiemer's disease where amyloid beta (Ab) oligomers and aggregates provide the apoptotic stimuli (Picone et al. 2009; Di Carlo 2010) . In addition to apoptosis, death receptor activation has been shown to be involved in other forms of cell death such as necrosis and necroptosis (Kim et al. 2000; Daniels et al. 2005) .. Death ligand induction leads to caspase 8 activation and thus could activate both type 1 and type 2 pathways. Understanding the mechanisms of type 1/type 2 regulation remains an important problem in the biology of apoptosis. Some of the initial experiments classified cells as either type 1 or type 2 depending on their mode of activation (Scaffidi et al. 1998 (Scaffidi et al. , 1999 . It was indicated that DISC formation and caspase 8 activation is stronger in type 1 cells (Scaffidi et al. 1998) . Even though initial studies found that increased caspase 8 activation can switch the activation from type 2 to type 1 (Scaffidi et al. 1998; Hua et al. 2005 ), a clear mechanistic understanding of such a correlation was lacking as caspase 8 molecules activate both type 1 and type 2 pathways (Okazaki et al. 2008) . Crucial insight into the problem can be provided by considering the kinetic rate constants for reactions involving caspase 8 with its immediate binding partners in type 1 and type 2 pathways. Preferential binding of capsase 8 with Bid (type 2 pathway) explains the type 2 phenotype in cells exhibiting low caspase 8 activation (Raychaudhuri et al. 2008; Raychaudhuri and Raychaudhuri 2013) . The levels of initiator caspases (pro-caspase 8 and pro-caspase 9) have also been shown to be key regulators of the type 1/type 2 choice (Okazaki et al. 2008) . In a previous work, we have shown that type 1/type 2 choice is regulated at a systems level . How fast the activation signal propagates through the type 1 and type 2 pathways, to finally activate effector caspases (caspase 3/7), was shown to determine the type 1/type 2 choice. Therefore, it becomes clear that both caspase 8 activation and systems level regulation downstream of caspase 8 activation determine the type 1/type 2 choice. Even though an important regulatory role of active caspase 8 in the type 1/type 2 choice has been established, the mechanism of caspase 8 activation (such as by death ligand induction) remains to be clearly elucidated. In this context, one may ask whether concentration of death ligands can impact the type 1/type 2 choice in apoptosis signaling. One also needs to consider the effect of inherent state of the membrane proximal module on the type 1/type 2 choice. Due to its crucial role in the regulatory mechanisms of apoptosis signaling, the question of type 1/type 2 choice has important ramifications for cancer and neurodegenerative disorders. We have recently studied the possibility of switching the activation from type 2 to type 1 by targeting the membrane proximal apoptotic signaling module in cancer cells . Recent experiments also explored how a type 2 to type 1 transition can be achieved in the context of TRAIL induced apoptosis (Kurita et al. 2011) . The problem of type 1/type 2 choice also has important ramifications for neurodegenerative disorders such as in Alzheimer's disease. Aggregated form of Ab peptides were shown to activate the type 1 pathway of apoptosis, whereas Ab oligomers mainly activated the type 2 pathway (Picone et al. 2009 ). Caspase 8 activation was observed under the action of both Ab oligomers and aggregates (even though Ab oligomers may enter the cytosol and mediate its dominant apoptotic effect). Therefore, a mechanistic understanding of the activation of the membrane proximal signaling module and its role in the type 1/type 2 choice, during apoptotic activation, can have profound implications for therapies for cancer and neurodegenerative disorders.. Activation of the membrane proximal apoptotic module, hence the type 1/type 2 choice, is further impacted by a systems level feedback mechanism mediated by caspase 6. It is known that active caspase 3 can proteolytically cleave pro-caspase 6 and generate active caspase 6 molecules. Caspase 6 has effector activities (similar to other effector caspases such as caspase 3/7) but they are also capable of activating caspase 8. Thus caspase 8 activation, upon death ligand induction, can propagate the activation signal through the type 1/type 2 loop network finally leading to activation of caspase 3 ? caspase 6 ? caspase 8 in a systems level feedback loop (Fig. 1) . Here, we study the effect of caspase 6 mediated feedback activation and elucidate the role of caspase 6 on the type 1/type 2 choice. Caspase 6 has attracted much recent attention in the context of neural cell apoptosis and neurodegenerative disorders (Leblanc 2013) . However, activation of caspase 6 in some of these cases might be through non-canonical pathways (different from the usual caspase 8 ? caspase 3 ? caspase 6 activation).. Cell-to-cell variability in apoptosis signaling has been established as an important regulatory mechanism in apoptotic cell death (Raychaudhuri et al. 2008; Raychaudhuri 2010; Albeck et al. 2008; Dussmann et al. 2010; Lee et al. 2010) . Expression levels of pro-and anti-apoptotic proteins are usually cell-type specific (Lopez-Araiza et al. 2006 ), but significant cell-to-cell variability (within a population of a given cell type) in protein levels has been observed (Spencer et al. 2009 ). Such cellular variability in apoptotic molecules can generate single cell variability in apoptotic activation (Spencer et al. 2009; Skommer et al. 2010) . However, even when all the cellular parameters are identical, cell-to-cell variability can result from inherently stochastic signaling reactions (Raychaudhuri et al. 2008; Raychaudhuri 2010; Skommer et al. 2010 ). Interestingly, cell-to-cell inherent variability can have important implications for diseases in which apoptosis is dysregulated Skommer et al. 2010 Skommer et al. , 2011a Brittain et al. 2010) . In a recent work, we have shown that low probability activation of Bax by BH3 only activators (such as Bid) is a mechanism for generating cell-to-cell variability (Raychaudhuri and Das 2013) . We have also found that cell-to-cell variability in this situation is remarkably sensitive to the Bcl-2 to Bax ratio (Bcl2 to Bax ratio [1 was linked to slow apoptotic activation with cellto-cell stochastic variability). In cancer cells, high Bcl2 to Bax ratio is frequently observed and is a mechanism for apoptosis resistance of cancer cells (Certo et al. 2006) . In contrast, in Alzheimer's diseases a recent study has shown that the Bcl2 to Bax ratio (to \ 1) can be altered by Ab peptides to favor apoptotic activation, which can then be reversed by the application of brain-derived neurotrophic factors (BDNF) to protect neural cells (Sun et al. 2012) . Therefore, novel chemotherapeutic strategies (in cancer and neurodegenerative disorders) might be possible to design that would rely on modulating cell-to-cell stochastic variability in apoptotic activation. Our previous works have indicated that the problem of type 1/type 2 choice is linked to the issue of cell-to-cell stochastic variability in apoptosis (as activation through the type 2 pathway is frequently associated with stochastic variability while type 1 activation remains deterministic) (Raychaudhuri et al. 2008; Raychaudhuri and Raychaudhuri 2013) . Hence, elucidating the role of death ligands and the membrane proximal signaling module in the type 1/type 2 choice can have major implications for understanding cell-to-cell stochastic variability in apoptosis activation and also for apoptosis related diseases.. In this work, we develop a hybrid Monte Carlo (MC) simulation to study the question of type 1/type 2 choice in apoptosis activation. Specifically, we address how the following parameters impact the type 1/type 2 choice: (1) concentration of death ligands and (2) the inherent state of the membrane proximal module. Our results indicate increased death ligand concentration increases type 1 activation and may lead to a type 2 to type 1 transition for a given cell type. The expression levels of pro-and antiapoptotic proteins in the membrane proximal signaling module are also shown to be crucial in caspase 8 activation and the type 1/type 2 choice. Our study provides mechanistic insights into the process of DISC clustering and caspase 8 activation (upon induction of death ligands). In addition, we consider the systems level feedback Fig. 1 mechanism mediated by caspase 6 activation and elucidates its role on type 1/type 2 choice. We discuss the implications of our results for cell-to-cell stochastic variability in apoptotic activation and show how such stochastic variability can be utilized to design therapies for cancer and neurodegenerative disorders. In this work, we elucidate that the concentration of death ligand and the inherent state of the membrane proximal signaling module affect the type 1/type 2 choice in apoptotic activation. We demonstrate that the inherent state of the membrane proximal module, such as the expression level of death receptors, can make cells of a given type prone to either type 1 or type 2 activation. However, concentration of the death ligand also emerges to be a key parameter in the type 1/type2 choice. Increased concentration of death ligand induction increases type 1 activation and may lead to a type 2 to type 1 transition in a given cell type. Our results provide a mechanistic understanding of caspase 6 activation mediated systems level feedback regulation (through caspase 8 activation) and its impact on the type 1/type 2 choice. Results obtained in this study also indicate an important role of cell-to-cell stochastic variability in caspase 8 activation. Such stochastic variability cannot be captured by ODE based models of caspase 8 activation. Majority of previous studies exploring the type 1/type 2 choice in apoptosis did not consider the effect of cell-to-cell variability in apoptotic activation.. In this study, formation of DISC and capsase 8 activation was modeled in a simplified manner. Additional complexities might arise from variation in the mode of ligand presentation (such as valency) (Huang et al. 1999) , explicit simulation of decoy receptors (Pan et al. 1997 ) and lipid mediated interactions (George and Wu 2012; Song et al. 2007 lead to more robust DISC formation and induction of type 1 activation in type 2 cells such as hepatocytes (Huang et al. 1999) . Taken together with our results, strength of death ligand mediated apoptotic stimuli seems to be a crucial regulator of the type 1/type 2 choice, irrespective of cell types (Raychaudhuri 2010) . In this context, it should be noted that membrane bound death ligands might be more effective in inducing receptor clustering in the restricted geometry of the 2-dimensional cell-cell contact. Lipid mediated interactions have been implicated in death ligand induced apoptotic activation even though a complete mechanistic understanding is lacking. Some studies indicate that Fas receptors are localized in sphingolipid rich regions of the plasma membrane in type 1 cells (Sanlioglu et al. 2005 ). In our current model, clustering of ligand bound death receptors was modeled in an effective manner and is governed by a free energy parameter (E dd ). Explicit simulation of lipid-mediated interactions would allow us to simulate novel strategies to target the membrane proximal signaling modules in cancer cells (Gajate and Mollinedo 2011; Song et al. 2007; Thome et al. 2012; Xiao et al. 2011; Gulbins and Kolesnick 2003) .. We considered the effect of cell-type specific variations in the membrane module on death ligand induced apoptotic activation. However, cell-to-cell variations in the expression levels of signaling molecules [such as the level of death receptors (Meng et al. 2011) ] can be significant even within a specific cell type. Such cellular variability may arise from epigenetic regulations and stochastic gene expressions. Based on our results it can be inferred that cell-to-cell variations in the membrane module for a given cell population (of the same type) should also impact the type 1/type 2 choice in apoptotic activation. Therefore, even within a cell population of the same type, one cell equipped with high expression level of death receptor and membrane proximal pro-apoptotic molecules may activate the type 1 pathway whereas another cell having small amounts of death receptor and other pro-apoptotic membrane molecules might exhibit the type 2 signaling phenotype. Interestingly, such type 1/type 2 variations at the level of single cells can also result from inherent stochastic variability, even when all the cellular parameters (such as the concentrations of membrane molecules) remain identical. Cell-to-cell variability in the time-to-caspase 8 activation (resulting from inherent stochastic fluctuations), for example, can impact the type 1/type 2 choice at the level of single cells. The regulation of type 1/type 2 choice at the level of single cells is an unexpected feature that emerged from stochastic Monte Carlo modeling.. Cell-to-cell stochastic variability has emerged as a key systems level regulatory mechanism in type 2 apoptosis. Thus the question of type 1/type 2 choice, in apoptotic activation, is frequently linked to the choice between stochastic (large cell-to-cell variability) and deterministic activation. In this work, we elucidated that the inherent state of membrane proximal signaling module and concentration of death ligand affect cell-to-cell variability in apoptotic activation. How cell-to-cell variability can be utilized to target the apoptotic pathway in treatment of cancer and neurodegenerative disorders remains to be explored. In our previous studies, we have explored potential options for inducing stochastic to deterministic transition (or at least minimize stochastic variability), selectively in cancer cells (Skommer et al. 2011a; Raychaudhuri and Das 2013; Raychaudhuri and Raychaudhuri 2013) . It was also indicated that cell-to-cell inherent variability should help protect normal cells during cancer chemotherapy. In a similar manner, strategies based on utilization of cell-to-cell variability might be effective in protecting cells from apoptosis in neurodegenerative disorders.. Application of death ligands (such as TRAIL), to induce apoptotic activation in cancer cells, has been established as a chemotherapeutic strategy (Picarda et al. 2012; Shirley et al. 2011; Kurita et al. 2011) . However, one needs to consider how to selectively target cancer cells and minimize fractional cell killing. In certain cancer cells, increased expression of death receptors and/or decreased amount of decoy receptor can enhance susceptibility of the cancer cell's membrane module to external apoptotic stimuli (Pan et al. 1997; Sheridan et al. 1997) . Such a mechanism presumably explains selective killing of cancer cells observed under TRAIL induction. Furthermore, vulnerability of the membrane module can be selectively induced (or increased) in some types of cancer cells, such as by increasing the death receptor expression by genetic mechanisms (Ho et al. 2010 ). However, anti-apoptotic molecules in the membrane proximal signaling module, such as cFLIP, are also frequently over-expressed in cancer cells imparting apoptosis resistance in those cancer cells (Safa and Pollok 2011) . In those cases, selective killing of cancer cells can be achieved by simultaneous targeting of a susceptible membrane module and inhibition of cFLIP. In this context, recombinant death ligands or monoclonal antibodies to death receptors (Picarda et al. 2012) , having low affinity for death receptors, can be utilized to increase specificity of targeting cancer cells. When we simulate apoptotic activation of susceptible cancer cells having increased death receptor expression (*100 molecules) but no cFLIP (also over-expressed Bcl2 family pro-and antiapoptotic proteins), under the application of small amount of death ligands (*2 molecules) with low affinity (*10 3 M -1 compared to 10 9 M -1 for typical death ligands), rapid type 1 dominant activation was observed (82 % cell death in 10 8 MC steps). In contrast, apoptotic activation in a normal cell with lower death receptor expression (*10 molecules) but significant cFLIP (*30 nM) expression was much lower (7 % cell death in 10 8 MC steps). In addition, the slow activation was dominated by the type 2 pathway with large cell-to-cell stochastic variability. There might exist other mechanisms of perturbing the susceptible membrane module of the apoptotic pathway, such as by combined targeting of death receptors and lipid rafts in a synergistic manner, to induce selective activation in cancer cells. Such selective targeting of the cancer cells can be achieved if only weak type 2 activation is induced in normal cells (where slow activation with large cell-to-cell variability should provide protection).. Caspase 6 activation has been frequently observed in neural cells undergoing degeneration and caspase 6 inhibition is emerging as a promising novel strategy in neurodegenerative disorders such as Alzheimer's disease (Leblanc 2013) . In this study of death ligand induced apoptosis, we considered systems level feedback regulation mediated by active caspase 6. Even if the apoptotic trigger is located in the intrinsic (type 2) pathway, caspase 6 mediated activation of caspase 8 can start direct processing of pro-caspase 3 (type 1 activation). Here, we showed how the activation can be switched from type 1 to type 2 under caspase 6 inhibition. Such a strategy, based on systems level regulatory mechanisms and cell-to-cell stochastic variability, could protect a large fraction of cells by minimally perturbing the apoptotic pathway and might turn out to be effective under certain situations. How regulation of cell-to-cell variability can be utilized to protect cells in neurodegenerative disorders remains to be explored. Recent studies indicated the possibility of caspase 6 selfactivation in the context of neurodegenerative disorders (Leblanc 2013) . However, over-pressed caspase 6 was thought to be required (for such self-processing) which would implicate low probability of activation and a role of stochastic variability (Raychaudhuri et al. 2008; Raychaudhuri and Das 2013) . In silico studies, such as the Monte Carlo simulations carried out in this work, can be utilized to design optimal strategies for targeting of the apoptotic pathway based on elucidation of its systems level regulatory mechanisms. Expression levels of death receptor and other molecules in the membrane proximal signaling module are key determinants of the type 1/type 2 choice in apoptosis Expression levels of death receptors (DRs), death adaptor proteins, apoptotic procaspase 8 and anti-apoptotic cFLIP define the inherent state of the membrane proximal signaling module. Death ligand induction perturbs the membrane proximal module leading to clustering of death receptors and adaptor proteins in the form of DISC and subsequent activation of caspase 8. We first consider the effect of variation in death receptor concentration on apoptotic activation keeping other molecules (including the ones in the membrane module) fixed. Death receptor expression is known to be cell type specific though cell-to-cell variability in death receptor level can be significant (Meng et al. 2011) . The celltype specific variability in death receptor expression has recently been shown to be a regulator of the type 1/type 2 choice in apoptotic death signaling (Meng et al. 2011) . Here, we explore the mechanistic basis of the observed correlation between death receptor expression and the type 1/type 2 choice in apoptosis signaling. We studied apoptotic activation as the number of death receptors was varied in the following manner: 2, 10 and 100 molecules. Death ligand concentration was kept constant at 10 molecules in these simulations. In Fig. 2 receptor expression (SKW 6.4 has *225 molecules/lm 2 and H9 has *190 molecules/lm 2 ) (Meng et al. 2011 ). In our simulation, type 1 activation was observed for lower concentration of death receptors (than found in typical type 1 cells), but we simulate an effective expression of death receptors and some of the anti-apoptotic factors, such as the expression of decoy receptors (Pan et al. 1997) , are not explicitly considered. In addition, lipid mediated interactions and other complexities of DISC formation may modify the type 1/type 2 choice. However, increased type 1 activation with increasing expression of death receptor levels, as observed in our in silico studies, should be robust. In our simulations, increased death receptor expression led to more rapid and increased generation of active caspase 8 molecules ( Fig. 2) and thereby increased type 1 signaling. Previous studies including that carried out in this lab have indicated a key role of active caspase 8 molecules in the type 1/type 2 choice (Scaffidi et al. 1998; Hua et al. 2005; Raychaudhuri et al. 2008; Raychaudhuri and Raychaudhuri 2013) . However, strong caspase 8 activation may not always lead to type 1 activation as, for example, type 2 signaling has been observed in caspase 3 deficient MCF-7 cell line despite significant active caspase 8 generation in those cells (Scaffidi et al. 1998) .. Consistent with our earlier findings (Raychaudhuri et al. 2008; Raychaudhuri and Raychaudhuri 2013) , type 2 activation was accompanied by large cell-to-cell stochastic variability (Fig. 3a) . Interestingly, cell-to-cell variation in the caspase 8 activation (within a cell population) provides a mechanism for generating cell-to-cell stochastic variability in apoptosis, especially when the death receptor concentration is low (*2 molecules) (Fig. 3b) . Clearly, cell-to-cell variability in caspase 8 activation contributes to cell-to-cell variability in effector caspase activation. In Fig. 3c , we show that the time to initial caspase 3 (10 % of total) activation, at the level of single cells, is correlated with the initiation of caspase 8 activation.. It is reasonable to expect that the inherent state of the membrane proximal signaling module is cell type specific and combined effect of all the molecules in the membrane module impact the type 1/type 2 choice. In our simulations, we varied the number of molecules in the membrane proximal signaling module in the following manner: (1) FADD = 10, cFLIP = 10 and procaspase 8 = 10, (2) FADD = 100, cFLIP = 100 and procaspase 8 = 100 and (3) FADD = 100, cFLIP = 10 and procaspase 8 = 100. Concentrations of both death ligands and death receptors were kept constant at 10 molecules. In our simulations, FADD represents the adaptor proteins that bind to both death receptor and intracellular signaling molecules such as procaspase 8 (see ''Methods'' section). In Fig. 4 , we show the type 1 fraction of activation as the membrane proximal signaling module is varied. Increased type 1 activation correlated well with increased DISC formation and generation of active caspase 8 molecules (Scaffidi et al. 1998) . The timeto-death decreased with increasing type 1 activation: T d = 4.3 9 10 7 MC steps for FADD = 10, cFLIP = 10, procaspase 8 = 10; T d = 3.9 9 10 7 MC steps or FADD = 100, cFLIP = 100, procaspase 8 = 100 and T d = 1.4 9 10 7 MC steps for FADD = 100, cFLIP = 10, procaspase 8 = 100. In our simulations, increased cFLIP level (or the cFLIP/ procaspase 8 ratio) led to reduced level of active caspase 8 generation resulting in decreased apoptosis and increased type 2 signaling. When we simulated Jurkat T cell (leukemia cell line) parameters obtained in (Hua et al. 2005) , type 2 signaling dominated apoptotic activation. However, when cFLIP level was inhibited, there was marked increase in type 1 activation. Increased cFLIP level has been observed in certain cancer cells and also been correlated with cancer chemoresistance (Safa and Pollok 2011 ). However, when cFLIP level was low but FADD and procaspase 8 levels were high, fast and robust type 1 activation was observed in our simulations (Fig. 4) . Thus it might be possible to induce strong apoptotic activation in cancer cells, by application of death ligands, when cFLIP level is low (or inhibited) but other molecules in the membrane module are robustly expressed.. Increase in death ligand concentration increases type 1 activation and crucially regulates the type 1/type 2 choice Concentration of death ligands presumably varies in vivo and the effect of such concentration variation needs to be studied in a systematic manner. We carried out controlled in silico experiments where the concentration of death ligands was varied keeping all other parameters constant. Death ligand concentration was varied in the following manner: 2, 10 and 100 molecules. For each of these death ligand concentrations the level of death receptor was varied in a similar manner: 2, 10 and 100 molecules. We observed that increased level of death ligand induction led to enhanced type 1 activation (Fig. 5) . When both death ligand and death receptor concentrations were high, rapid deterministic activation of the type 1 pathway has been observed. However, for low level of death ligand induction (*1-2 molecules), apoptotic activation was dominated by the type 2 pathway with large cell-to-cell variability (Fig. 6) . Therefore, even in cells that are known as type 1 cells (equipped with high level of death receptor expression) it might be possible to alter the signaling phenotype from type 1 to type 2 by decreasing the death ligand concentration [Especially when the interaction between death domains of the death receptor and the adaptor molecule is high affinity with K d *10 nM (Miyazaki and Reed 2001) ]. In a similar manner, large amount of death ligand induction might increase type 1 activation in type 2 cells. However, in type 2 cells having very low expression of death receptors, such as hepatocytes (Meng et al. 2011) , it becomes difficult to activate the type 1 pathway and activation of a second initiator caspase (caspase 9) becomes essential for robust activation of effector caspases. For type 2 cells equipped with very low death receptor expression it might still be possible to activate the type 1 pathway in the following manner: (1) when the type 2 pathway is blocked resulting in a slow type 1 activation, (2) by enhancing expression of pro-apoptotic membrane molecules especially the death receptor expression (Meng et al. 2011), (3) by inhibiting anti-apoptotic proteins such as cFLIP or XIAP (Jost et al. 2009; Raychaudhuri and Raychaudhuri 2013) . Other cell-type specific mechanisms for type 2 to type 1 transition are possible, such as, removing sequestration of c-MET bound Fas receptors by application of HGF (hepatocyte growth factors) (Accordi et al. 2007) or enhancing the expression of caspase 3 in caspase 3 deficient cell line MCF-7 (Scaffidi et al. 1998 ). Therefore, it seems reasonable to expect that both type 1 and type 2 pathways can get activated irrespective of cell types.. Our results indicate that death ligand concentration and the inherent state of the membrane module would regulate the clustering of DISC and thereby govern caspase 8 activation. The parameter (E dd ) governing the free-energy reduction of two neighboring death-ligand bound receptors should also be a regulator of DISC generation. Varying this free-energy parameter, resulted in altered clustering of death receptors and DISC generation. The effect of death ligand concentration on the type 1/type 2 choice is frequently mediated by variation in death receptor clustering and DISC generation. Increased clustering was observed in the case of high death ligand level (100 molecules), which seems to correlate well with activation of caspase 8 (Supplemental Fig. 1 ). The effect of increased death ligand concentration on receptor clustering was more pronounced when the free energy parameter E dd = -3 K B T (Supplemental Fig. 1b) . As mentioned earlier (''Methods'' section), E dd is an effective parameter and may vary depending on the cell type. In addition, it might be possible to enhance E dd (Legembre et al. 2005; Thome et al. 2012) selectively in cancer cells and induce apoptosis by death ligand induction or generating DISC formation by some other mechanisms.. Caspase 6 provides a system level feedback loop for apoptotic pathways and thereby impacts the type 1/type 2 choice Caspase 8 activation initiates signaling through the type 1 and type 2 pathways ultimately resulting in activation of effector caspases (caspase 3/7), thus creating a loop network structure at the systems level. Caspase 6 is another effector caspase that is activated by active caspase 3, but once activated it could also activate caspase 8 providing a mechanism for systems level feedback regulation. It is expected that a significant amount of active caspase 3 will be utilized to carry out their effector functions and only a fraction of it will be available for processing pro-caspase 6. We do not explicitly model the effector activities of active caspase 3, instead, an effective rate constant is assumed for caspase 3 activation of caspase 6. In our simulations, the rate constants for binding (P on : between caspase 3 and procaspase 6) and catalytic activation (P cat : generation of active caspase 6 from caspase3-pro-caspase 6 complex) were taken to be 0.1 times that was used for caspase 3 activation by caspase 9. Similar rate constants were assumed for binding and catalytic activation for caspase 6-pro-caspase 8 reaction. It is reasonable to expect that, similar to active caspase3, a significant portion of active caspase 6 will engage its effector substrates (such as lamin A) resulting in effective lower rate constants for its activation of caspase 8. Caspase 6 activation led to increased generation of active caspase 8 molecules and thus favored the type 1 activation. In Fig. 7a , we show the type 1 fraction of activation for two different combinations of death ligand and death receptor concentrations, with and without caspase 6. When both death ligand and death receptor concentrations were moderate (*10 molecules or higher), rapid generation of active caspase 8 molecules led to fast deterministic activation of the type 1 pathway and the effect of caspase 6 was to slightly reinforce the fast type 1 activation. For low levels of death ligand concentrations (*2 molecules), caspase 6 mediated feedback loop assisted in enhancing type 1 activation. For such low death ligand concentrations, especially when death receptor concentration is also low, apoptotic activation was dominated by the type 2 pathway with large cell-to-cell variability and slow activation of caspase 3; Caspase 6 activation was also slow in accordance with slow activation of Caspase 3. Time-course of caspase 6 activation is provided in Fig. 7b . At the level of single cells, caspase 6 activation was prominent only when robust caspase 3 activation was achieved after apoptosome formation and caspase 9 activation. Thus Caspase 6 activation led to strong caspase 8 activation only after capsase 3 activation reached a significant level (at *5 9 10 7 -10 8 MC steps) (Fig. 7b, c) . Consistent with our findings, robust activation of both caspase 8 and caspase 3 has been shown to be delayed (*hour) in type 2 cells (Scaffidi et al. 1998 ) (though in those experiments apoptotic activation was probed at the population level). In some of our simulations, such as when death ligand *2 molec. and death receptor *100 molec., dominant mode of activation was altered from type 2 to type 1 due to caspase 6 activation (type 1 fraction: 0.37 ± 0.22 for caspase 6 = 0 and type 1 fraction: 0.52 ± 0.32 for caspase 6 = 30 molec.). Thus caspase 6 inhibition can be used as a strategy to switch the activation from type 1 to type 2 and to increase cell-to-cell variability. Such a strategy can have important ramifications for the recently explored role of caspase 6 inhibition in treatment of neurodegenerative disorders (Leblanc 2013) . The signaling model for apoptotic cell death. We carried out a computational study of death ligand induced apoptosis using Monte Carlo (MC) simulations. A simplified network model of apoptotic death signaling ( Fig. 1 ) was studied utilizing controlled in silico experiments. Both type 1 and type 2 pathways were simulated. In our model, casapse 8 (initiator caspase) activation initiates signaling through both type 1 and type 2 pathways. Death ligands (such as FasL/TRAIL) undergo binding/ unbinding reactions with death receptors with nanomolar affinity. Probability parameters P on and P off are used to simulate binding/unbinding between death ligands and receptors (those presumably vary depending on the receptor type). Ligand binding supposedly induces a conformational change in the death receptors leading to their oligomerization (Scott et al. 2009 ). In our model, two ligand bound death receptors reduce their free energy (E dd is taken to be -2 K B T unless specified otherwise) when occupying neighboring lattice sites and thus induce clustering of death receptors. Adaptor proteins (such as FADD/ TRADD for Fas receptors) could bind to ligand bound death receptors, with moderately high 10 8 M -1 affinity, using an interaction between their death domains (DD). Affinity for this receptor-adaptor interaction was assumed to be similar to that between the adaptor molecule and procaspase 8. Adaptor molecules could also bind to free death receptors albeit with a low probability (affinity *1 M -1 ) so that significant apoptotic activation does not occur without ligand induction. Death effector domains (DED) of the adaptor molecules are known to bind with the DED domains of pro-caspase 8 molecules (k on : 3.5 9 10 6 M -1 s -1 and k off : 0.018 s -1 ) (Hua et al. 2005 ). However, this interaction takes place only when adaptors are already bound to death receptors. Therefore, ligand binding could induce clustering of death receptors and recruitment of adaptor molecules to clustered death receptors. Procaspase 8 molecules are then recruited to the clustered adaptor proteins to generate the assembly of DISC, ultimately leading to active caspase 8 molecules through autoprocessing (Peter and Krammer 2003) . In this work, DISC formation was simulated using a hybrid simulation scheme between kinetic Monte Carlo model for signaling reactions with an explicit free energy based model for the clustering of death receptors (Raychaudhuri 2013) .. Active caspase 8 initiates signaling through both type 1 and type 2 pathways. In the type 1 pathway, caspase 8 directly processes procaspase 3 (effector caspases) to generate active caspase 3. In the type 2 pathway, caspase 8 cleaves Bid to an active form (tBid) which translocates to mitochondria to bind with Bax. We have earlier shown that higher affinity of active caspase 8 for Bid, compared with that for pro-caspase 3, leads to preferential activation of the type 2 pathway (at early times) . Additional complexities, in the initial type 1/type 2 choice, might arise from cell type specific rate of Bid cleavage (such as due to Bid phosphorylation status in a given cell type) (Ozoren and El-Deiry 2002) , but are not considered here. When two Bax molecules are bound to tBid (on the mitochondrial membrane) they could detach as an active Bax dimer. Apoptotic inhibitor Bcl2 molecules bind with tBid and Bax and thereby inhibit formation of active Bax dimers. It is also possible for Bid to directly activate Bax albeit with a low probability (Raychaudhuri and Das 2013) . Cytochrome c is released into the cytosol in an all-or-none manner when the number of active Bax dimers reaches a pre-assigned threshold value (Goldstein et al. 2000; Dussmann et al. 2010) . Cytochrome c release leads to cytochorme c-Apaf binding and the subsequent formation of multi-molecular cyto c-Apaf-ATP complex apoptosome. Formation of the apoptosome complex is modeled in a simplified manner (by a cyto c-Apaf-Apaf-cyto c complex) where Apaf represents the Apaf-ATP complex. In our Monte Carlo simulations, effective low probability of apoptosome formation generates stochastic variability in apoptotic activation, but once formed, induces rapid activation of downstream caspases 9 and 3 (Raychaudhuri et al. 2008; Lee et al. 2010; Bagci et al. 2006 ). Caspase 9 is an initiator caspase (similar to caspase 8), which gets activated in the clustered assembly of apoptosome (whereas caspase 8 gets activated in DISC). The CARD domain of pro-caspase 9 interacts with the Apaf CARD domain (Sheridan et al. 1997; Pan et al. 1997; Mantovani et al. 2001) . Active caspase 9, in turn, cleaves procaspase 3 to its active form caspase 3. Apoptotic inhibitor XIAP binds to procaspase 9, caspase 9, and caspase 3 using its BIR domains (Riedl et al. 2001; Shiozaki et al. 2003) , thus it could inhibit apoptotic activation in both type 1 and type 2 pathways. However, the release of mitochondrial Smac can antagonize the inhibition of XIAP (Huang et al. 2003; Sun et al. 2002) . Smac is released simultaneously with cytochrome c in an all-or-none type manner. Active Caspase 3 can process pro-caspase 6 to proteolytically active caspase 6, another effector caspase but it also has the ability to activate pro-caspase 8 (Sun et al. 2002) . Thus caspase 6 provides a systems level feedback loop for both type1 and type 2 apoptotic activation. The details of all the reaction moves that are downstream of caspase 8 and considered in our current simulation (of the apoptosis pathway) are provided in our earlier works . Activation of caspase 3 (effector caspase) closes the type 1/type 2 activation loop (that is initiated by active caspase 8) and thus can be taken to be a downstream readout of apoptotic cell death signaling; MC simulations were carried out to measure the time-course of caspase activation at a single cell level.. Even though the present model can capture some of the essential systems level regulatory mechanisms of the apoptotic signaling pathway, it has several simplifying assumptions. In our model, effects of functionally similar proteins are coarse-grained by a representative protein.. Bcl-2 (B cell lymphoma protein 2), for example, represents all the Bcl-2 family proteins (such as Bcl-2, Bcl-xL, Mcl-1) with similar anti-apoptotic properties. In this context, it should be noted that the death receptors/ligands simulated here capture the effect of only one type of receptor/ligand (such as Fas/FasL or DR4/TRAIL), not the combined effect of all the death receptors/ligands. In our present study, DISC formation has been modeled in a simplified manner. Different isoforms of cFLIP have been implicated in differential regulation of apoptotic activation (Safa and Pollok 2011; Fricker et al. 2010) . A role of cleavage product of cFLIP (such as p43-FLIP) in NF-kB activation has also been studied using mathematical modeling (Neumann et al. 2010) . Here, we consider only the anti-apoptotic action of cFLIP (such as by cFLIPs that competitively binds with adaptor proteins in the DISC signalosome). In our study, the threshold of apoptotic activation, such as the threshold death ligand needed to trigger apoptosis, is governed by the level of apoptotic inhibitors such as cFLIP, Bcl2 and XIAP. In addition, the rate of synthesis and degradation of signaling molecules (such as caspases) can modify the threshold of activation (Gu et al. 2011 ) but is not considered here.. Monte Carlo simulation of cell death signaling Monte Carlo approach has been shown to capture some of the complexities of signaling reactions such as the effect of spatial heterogeneity (Raychaudhuri 2013) . Each run of our Monte Carlo simulation corresponds to apoptotic activation in a single cell, thus, Monte Carlo can capture cell-to-cell stochastic variability including inherent variability. We utilize a hybrid Monte Carlo simulation scheme that combines the following approaches: (1) a probabilistic rate constant based (implicit free energy) kinetic Monte Carlo simulation for various reaction moves such as diffusion, binding/unbinding and catalytic cleavage; (2) an explicit free-energy based model that captures clustering of ligandbound death receptors utilizing energy-function based diffusion moves. Both of the above approaches have been utilized in previous works from this lab (Raychaudhuri 2013) . At each Monte Carlo (MC) step molecules are randomly sampled N number of times, where N is the total number of molecules (either free or complexed) present in the system. Therefore, at each Monte Carlo (MC) step, one molecule is sampled (on average) once to allow for either diffusion or a reaction move. All the intracellular reaction moves (considered in our current simulations) and the corresponding kinetic rate constants can be found in our earlier works. Diffusion moves are carried out to one of the randomly chosen neighboring sites (4 for membrane bound molecules and 6 for cytosolic molecules) provided the condition of mutual physical exclusion is satisfied. In the kinetic Monte Carlo part of the simulation, once a diffusion/reaction move is randomly sampled it is accepted only if a randomly generated number in [0, 1] is less than the pre-defined probability constant for that particular move (otherwise the move is rejected). The detailed balance condition is satisfied through the ratio P on /P off , the probability constants for the binding and unbinding reactions, respectively (Raychaudhuri 2013 ). In the explicit free energy based part of the simulation, diffusion moves are accepted based on Metropolis criterion (P accept = min[1, exp(-DE/K B T)], where DE is the free energy difference between the new and the previous configuration (Newman and Barkema 1999) .

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A neuron model of stochastic resonance using rectangular pulse trains Abstract Stochastic resonance (SR) is the enhanced representation of a weak input signal by the addition of an optimal level of broadband noise to a nonlinear (threshold) system. Since its discovery in the 1980s the domain of input signals shown to be applicable to SR has greatly expanded, from strictly periodic inputs to now nearly any aperiodic forcing function. The perturbations (noise) used to generate SR have also expanded, from white noise to now colored noise or vibrational forcing. This study demonstrates that a new class of perturbations can achieve SR, namely, series of stochastically generated biphasic pulse trains. Using these pulse trains as 'noise' we show that a Hodgkin Huxley model neuron exhibits SR behavior when detecting weak input signals. This result is of particular interest to neuroscience because nearly all artificial neural stimulation is implemented with square current or voltage pulses rather than broadband noise, and this new method may facilitate the translation of the performance gains achievable through SR to neural prosthetics. Non-linear threshold systems transmit weak input signals more accurately when exposed to an optimal level of noise, a phenomenon called stochastic resonance (SR) (Gammaitoni et al. 1998) . Since neurons are essentially noisy, thresholded, information transfer systems, SR has found multiple applications in neuroscience (McDonnell and Abbott 2009) . For example, the nervous system appears to leverage the noise inherent in neural circuits to improve information transfer (Ma et al. 2006; Stein et al. 2005; Faisal et al. 2008) , and much current work seeks to explain the mechanisms by which this is accomplished. Another application lies in injecting external noise into the nervous system to create SR, thereby artificially enhancing neural information transfer. Advances on this front have the potential to restore sensitivity and functionality to the damaged or diseased nervous system where, for example, age related or diabetes-induced neuropathy may weaken sensation in the peripheral nervous system (Sumner et al. 2003; Mold et al. 2004 ). The present study is motivated by this second aim, and we present an approach to achieve SR that is conducive to artificial stimulation of the nervous system.. Numerous studies have demonstrated improved performance across multiple sensory modalities by artificially introducing an optimal level of noise, but little of this technology has been applied in neural prosthetics outside of the laboratory. For example, SR enhanced performance in photoreceptors (Bahar and Moss 2004) , cochlear implants and audition (Jaramillo and Wiesenfeld 1998; Chatterjee and Robert 2001) , mechanoreceptors (Collins et al. 1996; Douglass et al. 1993; Manjarrez et al. 2002; Richardson et al. 1998 ), muscle receptors (Fallon et al. 2004) , cardiac regulation (Hidaka et al. 2001) , muscle motor units (Kouzaki et al. 2012) , finger force production accuracy (Mendez-Balbuena et al. 2012) , vibrotactile sensation (Liu et al. 2002) , and balance in subjects with neurological deficit (Priplata et al. 2006) .. However, there is a disconnect between studies of SR and neural prosthetics: nearly all work on generating and tuning SR employs broadband noise while nearly all neural prosthetics use square pulses to activate the nervous system. In all of the mentioned in vivo studies, and in canonical theoretical studies as well (Hanggi et al. 1993; Benzi et al. 1981; Collins et al. 1995; Longtin 1993) , SR research was overwhelmingly focused on broadband noise, such as white noise (in fact, this is true of nearly all classical SR studies in the physical sciences, for reviews see (Wellens et al. 2004; Lindner et al. 2004) ). This is an obstacle to the implementation of SR in neural prosthetics because by far the most common and studied type of neural stimulation is highly structured pulse trains (Merrill et al. 2005) , where a time series of rectangular current or voltage pulses is delivered to the neural tissue. Trains of current pulses are much more suitable than broadband noise for application in neural stimulation because existing devices make pulses easy to implement and the vast literature dealing with electrical stimulation of neural tissue can be brought to bear on the mechanisms and effects of the stimulation. Moreover, this approach is likely to facilitate regulatory approval because a large number of neural stimulation devices that use square stimulation pulses are approved for use in humans (for example peripheral nerve stimulators, spinal cord stimulators and deep brain stimulators), and investigations into electrical stimulation safety focus nearly exclusively on square pulses (Ballestrasse et al. 1985; Agnew et al. 1986; Merrill et al. 2005) . Therefore, to advance the development of neural prosthetics that can leverage SR it is necessary to know whether pulse trains can elicit SR behavior in neurons, and how this compares to SR achieved with traditional broadband noise.. The present study demonstrates that charge balanced biphasic pulses delivered with random interpulse intervals and pulse durations elicit classical SR behavior in a computational Hodgkin Huxley model neuron (Hodgkin and Huxley 1952) . Moreover, with optimized parameters, the stochastic pulse trains can perform better than broadband noise. These results may facilitate the translation of laboratory studies of SR in the nervous system to neural prosthetics by enabling the use of pulse train perturbations to generate SR, rather than broadband noise. Laboratory studies over many years and spanning many sensory modalities have shown that artificial SR can enhance or restore human function, but these results have not been translated to assistive devices. In part, this is due to the broadband noise source that is universally used to generate SR, as opposed to trains of rectangular pulses common to nearly all neural stimulation studies and neural prosthetics (Merrill et al. 2005) . This study demonstrates (using a Hodgkin Huxley model neuron) that it is possible to induce and tune SR using classic square pulse trains. This advance is important as it initiates a straightforward path between laboratory results and potential translational neuroscience technology that leverages SR. This approach may encourage the development of neural prosthetics that enhance the body's native sensory systems, perhaps to offset neuropathy or other diseases (Hanggi 2002; McDonnell and Abbott 2009 ), a prospect that has been motivating work in neuroscience SR for at least two decades.. We showed that the spectral content of the noise appears to be the main determining factor in the character of SR behavior, rather than the time domain dynamics, even when those dynamics differ significantly between square pulses and random processes (Fig. 7 ). It appears possible, therefore, to manipulate the perturbation (noise) spectrum to control the peak SNR location along the perturbation intensity dimension, as well as the width of the peak. Moreover, these manipulations can be made using only square pulse waveforms, meaning they could be implemented using existing technology. The results also showed that the correlation time of broadband noise, a feature that has been investigated in SR studies, has a stochastic biphasic pulse train analogue in the lobe cutoff frequency, suggesting that the membrane filtering mechanisms that shape SR when using broadband noise play a similar role when SR is generated with the biphasic pulse trains.. The optimization of C ∼ 1 over the stochastic pulse train parameters revealed tradeoffs between high C ∼ 1 and low perturbation amplitude (Figs. 2a, 3b and c) . At short interpulse intervals and long pulse widths, C ∼ 1 approached 0.95φ , or 95 % of the fidelity achieved when representing noiseless threshold-level inputs. These parameters generated high duty cycle pulse trains that, in the limit, mimic a dc perturbation. Of course, a true dc offset would also achieve "stochastic" resonance in the model; but in practice dc current may corrode the stimulating electrodes and result in tissue damage. Other parameter combinations resulted in high, near-optimal C ∼ 1 values ('+' in Fig. 3b ), but achieved this performance at less than half of the RMS perturbation amplitude. To obtain the highest possible C ∼ 1 or near-optimal C ∼ 1 at greatly reduced RMS amplitude is a tradeoff whose resolution will be application specific.. It is important to estimate the relevance of a given C ∼ 1 value in terms of behavior or performance to put the results in context and to determine an appropriate compromise between RMS perturbation amplitude and C ∼ 1 . For example, Fig. 5a shows a high amplitude perturbation is not clear. An accurate answer likely requires an understanding of the way information is encoded in action potentials, how the sensory neuron in question interacts with local circuitry to process the information, and perhaps even how the sensory information is used by executive centers for decision making.. We turned to the results of two psychometric studies to estimate the behavioral significance of changes in C ∼ 1 . The first study showed that the distribution of average neural firing rates during a stimulus presentation has a receiver operator characteristic that matches the psychometric curve of a monkey performing a tactile discrimination task, suggesting that average firing rates of sensory-related neurons contribute to (or at least correlate with) sensory discrimination at a behavioral level (Hernandez et al. 2000) . Moreover, these firing rates changed linearly with stimulus intensity. The second study showed that classical SR enhances people's ability to detect accurately sub-threshold tactile impulses (Collins et al. 1997) of roughly the same shape as the "trapezoidal" pulses presented to the model neuron in the current study. These psychometric results indicate that tactile discrimination acuity can be enhanced by SR, and that the average firing rate of our model neuron can be used as a proxy for sensory discrimination. Therefore, the sub-threshold input signal to the model neuron can be viewed as a series of sub-threshold tactile perturbations, and the average firing rate of the neuron maps to a point on a psychometric curve indicating the expected classification of a tactile event. The input signal can then be binned into segments where it is either maximum or zero, and the average firing rate during those bins determines the neuron's classification: either an event was present or no event was present. We assume that the average firing rate during a threshold level stimulus (C ∼ 1 ¼ φ , 67 Hz, Fig. 1 ) or higher corresponds to always classifying a bin as having an event, a firing rate of zero corresponds to never classifying the bin as having an event, and the probability of classifying a bin as having an event between the extremes is the firing rate normalized by 67 Hz.. The simple linear relation between average firing rate and bin classification yields an estimate of the impact that changes in C ∼ 1 have on a behavioral outcome, and shows that a reduction of 0.02 C ∼ 1 from its peak results in a 5.3 % loss in discrimination accuracy (Fig. 9a) . In this figure the average percent correct curve is superimposed on the corresponding C ∼ 1 for a simulation where a stochastic biphasic pulse train was added to the sub-threshold input signal, and shows, as expected, that the peak C ∼ 1 corresponds to the best performance. This 5.3 % loss in accuracy represents a substantial amount, 15.6 % of the total range from the maximum of 83.8 % correct to chance levels of 50 %. Of course, because the performance and C ∼ 1 curves are not linear, a reduction in C ∼ 1 will result in different performance losses depending on the location in the curve. The maximum performance value of approximately 84 % correct classifications is in rough agreement with previous sub-threshold tactile discrimination behavioral studies that report between 70 and 76 % correct at peak performance, depending on subject (Collins et al. 1997) . The separation between the average firing rate during bins with (red) and without (black) a stimulus present is the determining factor in the shape of both the C ∼ 1 and performance curves (Fig. 9b) . At low perturbation amplitudes the neuron Fig. 9 A model of the relationship between C ∼ 1 and performance in a sensory discrimination task. a A sub-threshold 6.5 mA input signal (described in Fig. 1 ) and stochastic biphasic pulse train perturbations were applied to the neuron model and the resulting C ∼ 1 is plotted (green), where the solid line is the average and the shaded region is one standard deviation. The perturbations were generated following the trajectory in parameter space denoted by the white line in Fig. 3b , example trials of which are given in Fig. 2a . The input signal was divided into bins where a crest of the input signal was either present or absent, and the average firing rate determined the neuron's classification of each bin as either having or not having an event. The average percentage correct classifications for each 2,075 ms simulation were then calculated, and averaged over n=20 simulations for each point on the curve (blue). A reduction of 0.02 C ∼ 1 from peak C ∼ 1 results in a 5.3 % loss in classification accuracy. b For each simulation the average firing rate during bins with (red) and without (black) events are plotted. Peaks in performance and C ∼ 1 curves correspond to the largest separation between the two firing rate curves (44.9Hz) rarely fires, and classifies every bin as having no event, attaining 50 % accuracy because half the bins actually have no event. As the distance between the two curves grows it reaches a point of maximum distinguishability, corresponding to peak performance (8.6 mA RMS amplitude), after which point false positives begin occurring frequently because the firing rate increases during bins without events. This analysis provides an approximation of how the gain of C ∼ 1 due to SR translates into a measurable performance variable.. The broad potential applicability of SR in biology has motivated numerous studies into how systems can exploit SR, what signals are best suited to benefit from SR and even how the noise used to generate SR can be tuned to boost its effects (McDonnell and Abbott 2009) . In addition to optimizing the noise color for SR (Hanggi et al. 1993) , work has been done to show how incorporating deterministic perturbations with noise can supplement SR. Extended alternating subthreshold dc offsets have been shown effectively to reduce the barrier for a system to become excited, or switch to a new stable state, thereby enhancing SR (Mason et al. 2000) . High frequency (with respect to the signal to be detected) sinusoids have also been combined with noise to produce a reduction in the noise intensity that otherwise would have been needed to reach the optimal SNR (Ullner et al. 2003) . The present study builds on these ideas and demonstrates that perturbations comprised only of square pulses, which are ideally suited for bioengineering applications, can also be used to generate and tune SR. There is no need for extended offsets that can damage neurons and electrodes, and the flexibility of square pulses readily permits asymmetric charge-balanced pulse designs, bursts of pulses, pulses at irregular intervals and other stimulation strategies that are often used in neuroscience and bioengineering. This result promotes new ways to examine how the nervous system uses SR for signal detection and processing, and to exploit SR to restore function to the damaged or diseased nervous system.

O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O 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Automating Logic Transformations With Approximate SPFDs Abstract-During the very large scale integration design process, a synthesized design is often required to be modified in order to accommodate different goals. To preserve the engineering effort already invested, designers seek small logic structural transformations to achieve these logic restructuring goals. This paper proposes a systematic methodology to devise such transformations automatically. It first presents a simulation-based formulation to approximate sets of pairs of functions to be distinguished and avoid the memory/time explosion issue inherent with the original representation. Then, it uses this new data structure to devise the required transformations dynamically without the need of a static dictionary model. The methodology is applied to both combinational and sequential designs with transformations at a single or multiple locations. An extensive suite of experiments documents the benefits of the proposed methodology when compared to existing practices. In this paper, a simulation-based procedure for a new representation of SPFDs, namely aSPFDs, is first presented. The aSPFD is an approximation of the original SPFD as it only contains information that is explored by the simulation vectors. As a result, this technique can alleviate the memory/runtime issues that may be encountered by formal approaches. In addition, this paper proposes an aSPFD-based logic restructuring algorithm with SAT for both combinational and sequential designs. This technique can be used for a wide range of applications, such as logic optimization, debugging and when applying engineer changes. Experiments demonstrate that aSPFDs provide a powerful and dynamic method to restructure a logic design to a new set of specification. It is able to restructure designs at a location where other methods fail. Further empirical analysis confirms that the resolution of the transformation depends on the number of vectors used. A higher success rate can be achieved if more input vectors are provided but, at the same time, more memory/computation resources may be required. Another recent work proposed by Chang et al. [10] [11] [12] presents a similar technique to the one proposed here in the context of design debugging. This technique was developed independently and first published at the same time with part of the work presented in this paper. It uses a set of input vectors and the SAT-based technique described in [1] to generate the signature of a node. This is the bit-vector of logic values that the function of the node should evaluate to in order to rectify the error. Then, it identifies all pairs of 1-0 bits of the signature and ensures that, for each pair, at least one of the fanins of the node has a value transition as well. Such a pair is referred to as a pair of bits to be distinguished (PBD). In a sense, due to the use of bit-values by both PBDs and aSPFDs, these two methodologies share in common in terms of implementation and operation. However, aSPFDs are defined upon the concept of SPFDs, a theory that represents all minterm pairs. The work in [10] [11] [12] is utilizing don't care and care sets, as they always use signatures and these can only represent a special type of subset of SPFD edges, namely a clique of the bipartite graph of the SPFD [27] . If we force ourselves to use only signatures to process the information, we can't generate the more general SPFD that might be there because a minterm is either in the care onset, the care offset, or don't care set. Due to this reason, a variety of techniques developed for SPFDs by other research groups in the past decade can be inherited with ease to operate on aSPFDs, a fact not true for PBDs. Finally, the methodologies to manipulate aSPFDs illustrated in this paper are unique and tailored to operate within this context. However, in our use of the SPFD concepts in the current paper, we also only use effectively don't cares and not the full power of SPFDs, so there is no inherent superior efficiency between our methods and the one in [10] [11] [12] .

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Discretization Error Analysis and Adaptive Meshing Algorithms for Fluorescence Diffuse Optical Tomography: Part II Abstract-In the first part of this work, we analyze the effect of discretization on the accuracy of fluorescence diffuse optical tomography (FDOT). Our error analysis provides two new error estimates which present a direct relationship between the error in the reconstructed fluorophore concentration and the discretization of the forward and inverse problems. In this paper, based on these error estimates, we develop two new adaptive mesh generation algorithms for the numerical solutions of the forward and inverse problems in FDOT, with the objective of error reduction in the reconstructed optical images due to discretization while keeping the size of the discretized forward and inverse problems within the allowable limits. We present three-dimensional numerical simulations to demonstrate the improvements in accuracy, resolution and detectability of small heterogeneities in reconstructed images provided by the use of the new adaptive mesh generation algorithms. Finally, we compare our algorithms both analytically and numerically with the existing conventional adaptive mesh generation algorithms.Index Terms-Adaptive meshing algorithms, error analysis, fluorescence diffuse optical tomography. Following the analysis in [1] , in this work, we developed two new adaptive mesh generation algorithms for the numerical solutions of the forward and inverse problems in FDOT. These adaptive mesh generation algorithms aim to reduce the error in the reconstructed optical images while constraining the number of vertices in the adaptive meshes (i.e., the number of unknowns in the discretized forward and inverse problems) to a predefined number. The key feature that distinguishes these algorithms from the existing conventional adaptive mesh generation algorithms is the fact that they take into account the interdependence between the solutions of the forward and inverse problems, the source-detector configuration, the location of the fluorophore concentration with respect to the sources and detectors as well as the formulation of the inverse problem. In addition, we analytically showed that the conventional adaptive mesh generation algorithms relying on the interpolation error estimates may lead to higher error in the reconstructed images than expected.. We also showed that the computational complexity of the our adaptive mesh generation algorithms can be reduced from or to or , for the forward problem. For the inverse problem, it can be reduced from to . Hence, the cost of the adaptive mesh generation is negligible as compared to the cost of the matrix inversion involved in the numerical solutions of the forward and inverse problems on the generated adaptive meshes.. Our numerical experiments show that using the adaptive meshes generated by the conventional adaptive meshing method which do not include domain specific factors may lead to higher errors in reconstructed images, especially when the fluorophore concentration is small. However, given a limited number of unknowns in the discrete forward and inverse problems, our mesh generation algorithms can significantly improve the accuracy of the reconstructed images in terms of the improved detectability of small fluorescent targets as well as the improved image resolution, as compared with uniform and conventional adaptive meshing method.. From the perspective of mean-square-error estimation, the regularization parameters provide a trade-off between the bias and the variance of reconstructed images [28] , [29] . It is wellknown that the bias due to regularization can sometimes overwhelm the error due to other sources in the reconstructed image. Our discretization error estimates given in Theorems 1 and 2 are functions of the regularization parameters and are given with respect to the solution of the variational formulation of the inverse problem. In the development of adaptive meshing schemes, we assume that the appropriate regularization parameters for the variational formulation of the inverse problem are determined a priori and therefore keep these parameters fixed for different meshing schemes. However, since our error estimates involve regularization parameters, it is possible to study the interplay between the choice of regularization parameters and the mesh size and adaptively select both in an attempt to reduce the overall error in reconstructed images. We leave for the future the exploration of the interplay between the adaptive selection of regularization parameters and the mesh size.. In this paper, as well as in our previous publications [16] , [22] , we introduced an approach to develop discretization error estimates and adaptive meshing algorithms that take into account domain-specific factors as well as the interdependence of the solutions of the forward and inverse problems. This approach is not limited to single frequency FDOT imaging system, and can be extended to multiple frequency and time-domain optical imaging systems. Additionally, a similar approach can also be utilized in developing discretization error estimates and adaptive meshing algorithms for other inverse coefficient estimation problems involving PDEs, such as electrical impedance tomography and microwave tomography. However, specific discretization error estimates and adaptive meshing algorithms may be different in different inverse coefficient estimation problems depending on the particular PDE and the inverse problem formulation. Furthermore, our framework also provides an approach which can be used to analyze other types of errors involved in diffuse optical tomographic imaging. For example, the error due to Born approximation, the error due to background heterogeneities (incorrect assumptions about the background optical properties), and the error in reconstructed images due to variations in the source and detector magnitudes and positions. [16] where means the smaller of and , means the greater of and , and are the modified Bessel functions of the first and second kind, respectively [30] , and . Under the same conditions, an analytical solution for the adjoint problem (5) can be obtained in a similar way. The solution of the problem in 3-D can be derived in a similar manner [31] , [32] .

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