Robustness of the double well model in the sparse coding case.

Orange squares represent the storage capacity for the model with the learning rule defined in Eq (6) in the flat potential case (r1 = 10−3). Blue squares represent the storage capacity for the model with the learning rule defined in Eq (7) in the deep potential case (r1 = 0.5). Each point is averaged over 10 independent realizations, and the error bar represents the standard deviation. Other parameters are given as: r2 = 1, c = 1, N = 1, 500, f = 4log(N)/N.</p

Fig 1 -

(A) Sketch of the synaptic model. In the presence of external input, a synapse can stay in the same well (I), or jump into another well (II), depending on its current state and the amplitude of the input. C describes the distance between the two wells, while r1 characterizes its depth. The larger r1, the faster synapses decay towards the minima of the potential. (B) Overlap between the input presented at time v and its corresponding attractor state, mv, as a function of time elapsed since the presentation of this input u − v, for different values of r1. The solid lines represent the theoretical prediction and squares represent the simulation results with a network of size N = 30, 000 (mean and standard deviation computed over ten independent realizations). Dashed lines mark the storage capacity where mv vanishes. For all lines, the value of C is chosen to optimize storage capacity. Other parameters are: r2 = 1, r3 = 0, θ = 0, f = 0.5, c = 0.05.</p

Fig 4 -

(A) Comparison between the storage capacity of Markovian binary synapses and binarized double well synapses defined in Eq (12). The parameters of the double well model are r1 = 0.1, r2 = 1, r3 = 0, θ = 0, f = 0.5, cN = 2000. The width of the potential C is chosen to maximize storage capacity. The transition probabilities of binary Markovian model are chosen to match probabilities of switching wells in the double well model. (B) Comparison between the SNRs of Markovian synapses and binarized double well synapses, as a function of time elapsed since the presentation of a given pattern. Green curve: SNR of the double well model with C = C* calculated using Eq (40). Red line: SNR of the Markov model with the same transition probability. Black line: Upper bound of SNR for two-state Markov models. Dashed line: critical SNR below which the memories are no longer retrievable [28]. The intersection between the SNR curve and this dashed line determines the storage capacity of the network. (C) Cartoon of the temporal evolution of the distribution of synapses that undergo potentiation at t = 0, i.e. those for which just before potentiation (top) and after (bottom). The highlighted region denotes the region around the maximum of the potential where synapses that reside in one of the wells can make a transition to the other well (r2 = 1). Just before potentiation (t = 0), the distribution is symmetric. The distribution is then shifted up by r2, before decaying towards the respective wells. As the distribution becomes asymmetric near 0 (within the range of 0 ± r2), the next presented uncorrelated patterns will cause more probability mass to shift towards the positive side than towards the negative side, as demonstrated in the bottom figure. (D) Distribution of connections that have been potentiated at time t = 0, at later times t = 2 and t = 3. Black dots indicate ±r2 = ±1, while red dots indicate the minima of the potential, ±C = ±2.7. As demonstrated in (C), an uncorrelated pattern leads to a larger probability of switching from the low well to the high well than the opposite transition, resulting in an increase in SNR from t = 2 to t = 3, as seen in (B). At longer times, the distribution eventually returns to a symmetric distribution, leading to a decrease in SNR.</p

Fig 3 -

(A) Dependence of optimal potential width C* on network size N and potential depth r1. There is a critical value that separates two regimes: When , capacity is optimized for a single well potential, C* = 0. When , C* is non-zero, and increases as N increases and r1 decreases, indicating more pattern presentations are required to induce the switch between two stable states of synapses. Our analytical and numerical computation suggest that , as indicated by the boundary of the black region. (B) Scaling exponent between p and N as a function of r1, with r3 = 0. Solid red lines: theoretical prediction for optimal C. Dashed red line: Theoretical prediction for C = 0. Blue circles: Simulation results for optimal C (mean and standard deviation computed over ten independent realizations). For , p ∼ log(N) gives a = 0. When , the model with C = C* has a power law dependence on N (red solid line), while the model with C = 0 gives a = 0 (dashed red line). The exponent a decreases with r1 and converges to 0.5 in the large r1 limit. Insets show distributions of synaptic weights for representative examples. The distribution of Jijs goes from unimodal when C = 0, to bimodal for the optimal C and to quasi-discrete when r1 becomes large.</p

Dependence of storage capacity <i>p</i> on potential width <i>C</i> and network size <i>N</i>, for representative examples of the decay rate <i>r</i>₁.

(A) Dependence of p on C for . In the small r1 limit, the optimal potential width C* is zero (i.e., a single well potential), and p decreases weakly with C. The number of stored patterns increases logarithmically with network size, as demonstrated in (C). (B) Dependence of p on C for . The number of stored patterns p reaches its maximum at a nonzero value of C. The optimal storage capacity increases as a power law of N, as demonstrated in (D). (C) Storage capacity of the single-well potential model (C = 0) as a function of with network size N. The storage capacity decreases when r1 increases as indicated by Eq (10). (D) Storage capacity of the double well potential model (C = C*, full lines) as a function of network size N. When , the optimal storage capacity is much larger than the storage capacity of the single well potential model (dashed lines with the same color). The dashed-dotted black line represents for reference. Other parameters in this figure are r2 = 1, r3 = 0, θ = 0, f = 0.5, c = 0.05.</p

Storage capacity of the model with double well synapses in the sparse coding limit.

(A) Storage capacity of the network as a function of r1 with the learning rule defined in Eq (6) (B) Storage capacity of the network as a function of r1 with the learning rule defined in Eq (7) (C) p as a function of N in the flat potential case (r1 = 10−3) for the model with learning rule defined in Eq (6). Optimal p is plotted as a function of N in a semi-log plot. Dashed red line: p = alog(N/b), a = 396.5, b = 294.5, R2 = 0.9964. (D) p as a function of N in the deep potential case (r1 = 0.5) for the model with the learning rule defined in Eq (7). Dashed red line: p = a(N/log(N))b, a = 0.212, b = 1.91, R2 = 0.9927. In all panels, data points are averaged over 10 realizations, and error bars represent the standard deviation. C and θ are chosen to optimize for storage capacity. Other parameter are given as: r2 = 1, r3 = 0, c = 1, f = 4 log(N)/N.</p

Dependence of Storage Capacity on Noise Strength <i>r</i>₃ for different values of Potential Width <i>C</i>.

When C is small (e.g., C = 0, 1, 2, 3), the storage capacity p monotonically decreases with the noise strength r3. However, for larger values of C (e.g., C = 4, 5), the storage capacity first increases with noise intensity, reaches a peak and then decreases. Thus, noise facilitates the learning of new memories in such cases. Other parameters are given as N = 10, 000, r1 = 0.1, r2 = 1, θ = 0, f = 0.5, c = 0.05.</p

Table_1_Machine learning-based predictive models and drug prediction for schizophrenia in multiple programmed cell death patterns.XLSX

BackgroundSchizophrenia (SC) is one of the most common mental illnesses. However, the underlying genes that cause it and its effective treatments are unknown. Programmed cell death (PCD) is associated with many immune diseases and plays an important role in schizophrenia, which may be a diagnostic indicator of the disease.MethodsTwo groups as training and validation groups were chosen for schizophrenia datasets from the Gene Expression Omnibus Database (GEO). Furthermore, the PCD-related genes of the 12 patterns were extracted from databases such as KEGG. Limma analysis was performed for differentially expressed genes (DEG) identification and functional enrichment analysis. Machine learning was employed to identify minimum absolute contractions and select operator (LASSO) regression to determine candidate immune-related center genes, construct protein–protein interaction networks (PPI), establish artificial neural networks (ANN), and validate with consensus clustering (CC) analysis, then Receiver operating characteristic curve (ROC curve) was drawn for diagnosis of schizophrenia. Immune cell infiltration was developed to investigate immune cell dysregulation in schizophrenia, and finally, related drugs with candidate genes were collected via the Network analyst online platform.ResultsIn schizophrenia, 263 genes were crossed between DEG and PCD-related genes, and machine learning was used to select 42 candidate genes. Ten genes with the most significant differences were selected to establish a diagnostic prediction model by differential expression profiling. It was validated using artificial neural networks (ANN) and consensus clustering (CC), while ROC curves were plotted to assess diagnostic value. According to the findings, the predictive model had a high diagnostic value. Immune infiltration analysis revealed significant differences in Cytotoxic and NK cells in schizophrenia patients. Six candidate gene-related drugs were collected from the Network analyst online platform.ConclusionOur study systematically discovered 10 candidate hub genes (DPF2, ATG7, GSK3A, TFDP2, ACVR1, CX3CR1, AP4M1, DEPDC5, NR4A2, and IKBKB). A good diagnostic prediction model was obtained through comprehensive analysis in the training (AUC 0.91, CI 0.95–0.86) and validation group (AUC 0.94, CI 1.00–0.85). Furthermore, drugs that may be useful in the treatment of schizophrenia have been obtained (Valproic Acid, Epigallocatechin gallate).</p

Trade-off between capacity and robustness when varying potential depth <i>r</i>₁.

(A) The storage capacity decreases when the noise strength increases strongly depends on r1. Small values of r1 lead to large storage capacity, but storage is highly sensitive to noise. Large values of r1 lead to small capacity, but storage is highly robust. (B) The trade-off between p0 and R, where p0 is the storage capacity in the absence of noise (r3 = 0), and R quantifies robustness to noise, Eq (14). With deeper potentials, the storage capacity is smaller but storage is more robust. Here, C is optimized for each value of r1 and other parameters are given as: N = 10, 000, r2 = 1, θ = 0, f = 0.5, c = 0.05.</p

Table_2_Machine learning-based predictive models and drug prediction for schizophrenia in multiple programmed cell death patterns.XLSX

BackgroundSchizophrenia (SC) is one of the most common mental illnesses. However, the underlying genes that cause it and its effective treatments are unknown. Programmed cell death (PCD) is associated with many immune diseases and plays an important role in schizophrenia, which may be a diagnostic indicator of the disease.MethodsTwo groups as training and validation groups were chosen for schizophrenia datasets from the Gene Expression Omnibus Database (GEO). Furthermore, the PCD-related genes of the 12 patterns were extracted from databases such as KEGG. Limma analysis was performed for differentially expressed genes (DEG) identification and functional enrichment analysis. Machine learning was employed to identify minimum absolute contractions and select operator (LASSO) regression to determine candidate immune-related center genes, construct protein–protein interaction networks (PPI), establish artificial neural networks (ANN), and validate with consensus clustering (CC) analysis, then Receiver operating characteristic curve (ROC curve) was drawn for diagnosis of schizophrenia. Immune cell infiltration was developed to investigate immune cell dysregulation in schizophrenia, and finally, related drugs with candidate genes were collected via the Network analyst online platform.ResultsIn schizophrenia, 263 genes were crossed between DEG and PCD-related genes, and machine learning was used to select 42 candidate genes. Ten genes with the most significant differences were selected to establish a diagnostic prediction model by differential expression profiling. It was validated using artificial neural networks (ANN) and consensus clustering (CC), while ROC curves were plotted to assess diagnostic value. According to the findings, the predictive model had a high diagnostic value. Immune infiltration analysis revealed significant differences in Cytotoxic and NK cells in schizophrenia patients. Six candidate gene-related drugs were collected from the Network analyst online platform.ConclusionOur study systematically discovered 10 candidate hub genes (DPF2, ATG7, GSK3A, TFDP2, ACVR1, CX3CR1, AP4M1, DEPDC5, NR4A2, and IKBKB). A good diagnostic prediction model was obtained through comprehensive analysis in the training (AUC 0.91, CI 0.95–0.86) and validation group (AUC 0.94, CI 1.00–0.85). Furthermore, drugs that may be useful in the treatment of schizophrenia have been obtained (Valproic Acid, Epigallocatechin gallate).</p